Antilog calculator

Enter a logarithm value and a base. The calculator evaluates \(b^y\), where \(y\) is the log value and \(b\) is the base. For common logarithms, use base 10. For natural logarithms, choose base \(\mathrm{e}\). For binary logarithms, choose base 2.

What this page covers

This page is specifically about antilogs and inverse logarithms. If you need to calculate a logarithm from a known number, use the log calculator. If you need powers and exponent values directly, the exponent calculator is the related tool. This page focuses on going backward from a logarithm value to the original number.

What is an antilog?

An antilog, short for antilogarithm, is the inverse of a logarithm. A logarithm answers the question, "What exponent gives this number?" An antilog answers the reverse question, "What number is produced by this exponent?" If a logarithm compresses a number into an exponent, the antilog expands the exponent back into the original number.

In this relationship, \(b\) is the base, \(x\) is the original number, and \(y\) is the logarithm value. The antilog of \(y\) in base \(b\) is \(b^y\). This is why an antilog calculator is also called an inverse logarithm calculator.

For example, because \(\log_{10}(100)=2\), the antilog base 10 of 2 is 100. The logarithm says that 10 must be raised to the power 2 to produce 100. The antilog performs that power calculation directly.

This inverse relationship is the main idea. To move from a number to its logarithm, use a logarithm. To move from a logarithm back to the original number, use an antilog. The two operations undo one another when the same base is used.

Base 10 antilog

The base 10 antilog is the inverse of the common logarithm. Common logarithms use base 10 and are often written as \(\log(x)\) when the base is understood. The base 10 antilog of \(y\) is \(10^y\).

Base 10 antilogs are common in chemistry, sound intensity, scientific notation, and older logarithm table work. They are also useful whenever a logarithmic scale is built around powers of 10. A log value of 1 corresponds to 10. A log value of 2 corresponds to 100. A log value of 3 corresponds to 1000. Decimal log values produce numbers between powers of 10.

Decimal log values are handled the same way. For example, the antilog base 10 of 2.5 is \(10^{2.5}\). This can be written as \(10^2\cdot10^{0.5}=100\sqrt{10}\), which is approximately 316.227766. The exponent is not required to be a whole number.

If you are learning logarithms from the forward direction, review logarithms and logarithmic functions. This calculator page is the reverse side of that topic: it turns a logarithm value back into the original number.

Natural antilog and base \(e\)

The natural logarithm uses base \(\mathrm{e}\), where \(\mathrm{e}\approx2.718281828\). The natural logarithm is written as \(\ln(x)\). Its inverse is the exponential function \(\mathrm{e}^x\). Therefore, the natural antilog of \(y\) is \(\mathrm{e}^y\).

Natural antilogs appear frequently in calculus, continuous growth, compound interest, biology, probability, statistics, and differential equations. The base \(\mathrm{e}\) is special because it is the natural base for continuous change. For example, if \(\ln(x)=4\), then \(x=\mathrm{e}^4\approx54.598150\).

Students often confuse \(\log(x)\) and \(\ln(x)\). In many school contexts, \(\log(x)\) means base 10, while \(\ln(x)\) means base \(\mathrm{e}\). Some programming languages and advanced mathematics contexts may use different conventions, so always check the notation. In this calculator, you can choose base 10, base \(\mathrm{e}\), base 2, or a custom base.

Natural antilogs are strongly connected to exponential functions. If you want to study this family of functions more broadly, see exponential function formula and examples, exponential functions, and exponential growth and decay.

Base 2 antilog

The base 2 antilog is the inverse of the binary logarithm. If \(\log_2(x)=y\), then \(x=2^y\). Base 2 is important in computer science, information theory, binary data, algorithm analysis, memory sizes, and digital systems.

For whole-number log values, base 2 antilogs produce powers of two:

For base 2 logarithms specifically, use the log base 2 calculator when you need the logarithm, and this antilog calculator when you need the inverse. The two tools solve opposite questions.

Example: if a logarithmic measure says \(\log_2(x)=8\), then the original number is \(x=2^8=256\). If \(\log_2(x)=10\), then \(x=2^{10}=1024\). If \(y\) is not a whole number, the calculator still evaluates \(2^y\).

Worked antilog examples

Example 1: Common antilog

Find the antilog base 10 of 4.

The original number is 10000. This means \(\log_{10}(10000)=4\).

Example 2: Decimal common antilog

Find the antilog base 10 of 1.7.

The result is between 10 and 100 because the log value 1.7 lies between 1 and 2. This is a useful reasonableness check.

Example 3: Natural antilog

Solve \(\ln(x)=2.3\).

The natural antilog of 2.3 is approximately 9.974182. Substituting back, \(\ln(9.974182)\approx2.3\).

Example 4: Base 2 antilog

Find \(x\) if \(\log_2(x)=12\).

The original number is 4096. This is why powers of two appear frequently in computing and binary systems.

Example 5: Custom base

Find \(x\) if \(\log_5(x)=3.4\).

The calculator can evaluate this directly by using base 5 and log value 3.4.

Choosing the correct base before finding an antilog

The base is not a small detail in an antilog calculation. It controls the scale of the answer. The same logarithm value can produce completely different original numbers when the base changes. For example, the antilog of 3 in base 10 is \(10^3=1000\), but the antilog of 3 in base 2 is \(2^3=8\), and the natural antilog of 3 is \(\mathrm{e}^3\approx20.085537\). All three calculations use the same exponent, but they answer different questions because their bases are different.

A good habit is to write the original logarithm statement before typing anything into the calculator. If the problem says \(\log_{10}(x)=y\), the base is 10. If it says \(\ln(x)=y\), the base is \(\mathrm{e}\). If it says \(\log_2(x)=y\), the base is 2. If the expression is written as \(\log_b(x)=y\), the base is the value of \(b\). This short step prevents the most common error: using a common antilog when the question actually asks for a natural antilog, or using a natural antilog when the original logarithm was base 10.

The base also helps you predict the size of the result. When \(b>1\), positive log values give answers greater than 1, zero gives exactly 1, and negative log values give answers between 0 and 1. For example, \(10^2=100\), \(10^0=1\), and \(10^{-2}=0.01\). If your antilog answer violates this pattern, check the sign, the base, and the typed value. A reasonableness check is often faster than redoing the entire problem.

When \(0

If you are moving between logarithms and exponents for algebra practice, compare this calculator with the exponent calculator. The antilog calculator focuses on reversing a logarithmic statement, while an exponent calculator is useful whenever you simply need to evaluate a power such as \(a^n\).

Characteristic and mantissa in common antilogs

Older textbooks, log tables, and some examination solutions may split a common logarithm into two parts: the characteristic and the mantissa. This language is most often used with base 10 logarithms. The characteristic is the integer part of the logarithm, and the mantissa is the decimal part. For example, the number 2.4731 has characteristic 2 and mantissa 0.4731. The antilog can then be understood as:

This is useful because \(10^2\) sets the place value, while \(10^{0.4731}\) gives a number between 1 and 10. In this example, \(10^{0.4731}\approx2.9739\), so:

The characteristic tells you roughly how many digits the result should have. If the common log value is between 2 and 3, the antilog is between \(10^2\) and \(10^3\), so it is between 100 and 1000. This makes characteristic and mantissa a powerful mental check even when you use a calculator. If \(10^{2.4731}\) came out as 29.739 or 29739, the decimal point would be suspicious because the result should be between 100 and 1000.

For positive common logs, the rule is straightforward. If \(y=n+m\), where \(n\) is an integer and \(0\le m<1\), then:

Here \(10^n\) shifts the decimal place, and \(10^m\) supplies the significant digits. This is why many log-table methods first found the antilog of the mantissa, then used the characteristic to position the decimal point.

Negative common logs need extra care because the integer and decimal parts can be written in more than one way. For instance, \(-2.3010\) can be read directly as \(10^{-2.3010}\), but in older table notation it may be written as \(\bar{3}.6990\), meaning \(-3+0.6990\). The bar notation keeps the mantissa positive so it can be looked up in a standard table. The value is:

The practical lesson is that negative logarithms are not a warning sign. They usually indicate that the original number is between 0 and 1 when the base is greater than 1. Scientific measurements, probabilities, pH values, small concentrations, and error rates often create negative logarithms. The antilog simply converts those compressed log-scale values back into ordinary positive numbers.

Students who are learning logarithm laws may find it useful to review logarithms alongside antilogs. The logarithm turns multiplication into addition, while antilog turns the result back into multiplication on the original scale. That connection is covered in more depth in logarithms and logarithmic functions.

How antilog tables work

Before scientific calculators were common, students and engineers often used logarithm and antilogarithm tables. Although this calculator gives the answer instantly, the table method is still worth understanding because it explains why logarithms were historically so useful. Tables allowed people to replace difficult multiplication, division, powers, and roots with addition, subtraction, multiplication by a constant, and lookup values.

An antilog table for common logarithms usually lists values of \(10^m\), where \(m\) is the mantissa between 0 and 1. Since \(10^0=1\) and \(10^1=10\), every mantissa antilog lies between 1 and 10. After finding this value, the characteristic places the decimal point. For example, if the log value is 3.2468, the mantissa is 0.2468. A table would give \(10^{0.2468}\approx1.7657\). Since the characteristic is 3, the full antilog is:

The table does not need to store every large number. It stores only the mantissa part, then the reader uses place value. This is efficient because the mantissa controls the significant digits and the characteristic controls scale.

Suppose the log value is 0.8129. The antilog must be between \(10^0=1\) and \(10^1=10\). Looking up the mantissa 0.8129 gives approximately 6.500. Since the characteristic is 0, the result is about 6.500. Now suppose the log value is 4.8129. The mantissa is still 0.8129, so the significant digits are still about 6.500, but the characteristic is 4:

The same mantissa gives the same significant-digit pattern, but the characteristic changes the magnitude. This is also why common logarithms are closely related to scientific notation. A number written as \(a\times10^n\), where \(1\le a<10\), has a common logarithm of:

The \(n\) part is the characteristic, and \(\log_{10}(a)\) is the mantissa. When you take the antilog, you reverse that structure.

Modern calculators hide the table, but they follow the same mathematical relationship. The calculator evaluates \(b^y\) directly. For base 10 it evaluates \(10^y\); for natural logs it evaluates \(\mathrm{e}^y\); for base 2 it evaluates \(2^y\). Understanding tables gives you a better sense of why the answer has the size it does, instead of treating the calculator output as an isolated decimal.

Interpreting negative, fractional, and large antilog values

Many antilog questions use whole-number exponents at first, but real data often produces decimals, negative values, and very large magnitudes. Fractional log values are normal. If \(y=0.5\) and \(b=10\), then the antilog is \(10^{0.5}=\sqrt{10}\approx3.162277660\). If \(y=1.5\), then \(10^{1.5}=10\sqrt{10}\approx31.62277660\). A fractional exponent simply places the result between nearby whole powers.

For base 10, the interval check is especially simple. If \(1

Negative log values do not create negative results. Instead, they create reciprocals:

For example, \(10^{-4}=1/10^4=0.0001\). In words, a negative common logarithm tells you that the original number is smaller than 1 but still positive. This matters in chemistry, medicine, engineering tolerances, and any field that measures very small positive quantities.

Large log values can create very large antilogs. For example, \(10^{12}=1,000,000,000,000\). It is usually clearer to keep this as \(1\times10^{12}\) rather than write every zero. Likewise, \(10^{-12}=0.000000000001\) is clearer as \(1\times10^{-12}\). Scientific notation helps prevent counting-zero errors and keeps answers readable.

When the base is \(\mathrm{e}\), the same ideas apply, but the benchmark powers are different. \(\mathrm{e}^0=1\), \(\mathrm{e}^1\approx2.71828\), \(\mathrm{e}^2\approx7.38906\), and \(\mathrm{e}^3\approx20.08554\). If a natural log value is 2.4, the antilog should be between \(\mathrm{e}^2\) and \(\mathrm{e}^3\). This kind of estimate makes the calculator output easier to trust.

Precision, rounding, and reporting antilog answers

An antilog can amplify small changes in the log value, especially when the base is large or the log value is large. For example, \(10^{2.30}\approx199.526\), while \(10^{2.31}\approx204.174\). The log value changed by only 0.01, but the original-scale value changed by more than 4.6. This is why rounding the logarithm too early can change the final answer.

As a rule, keep extra digits while calculating and round only the final answer. If a problem asks for three significant figures, calculate with more than three figures internally, then round the final antilog. If a data table gives log values to four decimal places, do not round them to one or two decimal places before taking the antilog unless the question explicitly tells you to do so.

Use the context to choose the final format. A school algebra problem may accept six decimal places. A chemistry concentration may need scientific notation. A sound intensity ratio may need a rounded multiplier. A business or statistics prediction may need units and a sensible number of decimal places. The mathematical result is \(b^y\), but the reported answer should match the problem's real-world meaning.

There is also a difference between exact and approximate answers. Expressions such as \(10^2\), \(2^8\), and \(\mathrm{e}^0\) may simplify exactly. Expressions such as \(10^{1.7}\), \(\mathrm{e}^{2.3}\), and \(5^{3.4}\) usually require a decimal approximation. In a written solution, it is often best to show both:

The exact exponential form shows the operation, and the decimal approximation gives the practical value.

If your calculator displays a value in exponential notation such as \(3.42\mathrm{e}{-7}\), read it as \(3.42\times10^{-7}\). This is not an error. It is the calculator's compact way of writing a very small number. For large and small results, exponential notation is usually clearer than a long string of digits.

More applied examples with antilogs

Recovering a quantity from a pH value

If a solution has \(\mathrm{pH}=4.6\), then the hydrogen ion concentration is:

The negative exponent tells you the concentration is a small positive number. A lower pH produces a larger hydrogen ion concentration because the exponent \(-\mathrm{pH}\) becomes less negative.

Recovering an intensity ratio from decibels

If a sound level is 30 dB above a reference intensity, the intensity ratio is:

This means the intensity is 1000 times the reference intensity. The logarithmic decibel value is compact, while the antilog reveals the actual multiplicative ratio.

Undoing a natural log transformation

Suppose a model predicts \(\ln(x)=5.25\). To return to the original scale, calculate:

If \(x\) represents a measured quantity, the final answer should include the correct units. A log-scale prediction by itself is often not meaningful to readers until it is converted back with the natural antilog.

Solving a growth equation after taking logs

Consider a simplified model \(A=A_0\mathrm{e}^{kt}\). If algebra gives \(\ln(A/A_0)=1.2\), then:

The antilog shows that the quantity is about 3.32 times its starting value. This style of interpretation is common in exponential models, so it is useful to understand both the log equation and the original-scale result.

Checking an answer by taking the log again

After finding an antilog, you can check the answer by applying the original logarithm. If the calculator gives \(x\approx50.118723\) for \(10^{1.7}\), then:

This reverse check is especially helpful in multi-step algebra problems where the antilog is only one part of the solution. When the check returns the original log value, the base and exponent were handled correctly.

For logarithm-side calculations, use the log calculator. For simplification practice involving logarithm laws, the log reduction calculator can help with expressions before you convert back to the original scale.

Step-by-step solution patterns for antilog problems

Most antilog questions follow a small number of patterns. Once you recognize the pattern, the work becomes much more predictable. The calculator gives the numerical value, but the written method is still important for classwork, homework, and exam solutions because it shows why the answer is correct.

Pattern 1: The logarithm is already isolated

This is the simplest pattern. If the equation is already in the form \(\log_b(x)=y\), rewrite it as \(x=b^y\). For example:

The calculator can handle the decimal exponent, but your written solution should still show the conversion from logarithmic form to exponential form. This confirms that you used the antilog and did not multiply or divide by the base.

Pattern 2: The logarithm contains an expression

Sometimes the log is applied to an expression rather than a single variable. In that case, first convert from log form to exponential form, then solve the remaining algebra. For example:

The antilog gives the value of the expression inside the logarithm, not always the final value of \(x\). A common mistake is stopping at \(x+5=64\). The last algebra step matters.

Pattern 3: A coefficient is outside the logarithm

If a logarithm is multiplied by a number, isolate the logarithm before taking the antilog. For example:

Do not take the antilog of 12 immediately. The logarithm value is 3 after division by 4, so the antilog is \(10^3\), not \(10^{12}\).

Pattern 4: Natural logs in exponential models

Natural logarithms often appear after solving growth or decay equations. If the equation becomes \(\ln(x)=k\), the reverse step is \(x=\mathrm{e}^k\). For example:

This is sometimes called exponentiating both sides. It is the same inverse operation as taking the natural antilog.

Pattern 5: Log values from a table or model

In statistics and science, you may be given a log-scale result from a table, regression output, or transformed data set. The result might say something like \(\log_{10}(x)=1.92\) or \(\ln(x)=4.08\). Your job is not to solve an equation from scratch; it is to return the result to the original scale. For base 10, use \(10^y\). For natural logs, use \(\mathrm{e}^y\). Always state the original units after converting back, because the log-scale number and the original-scale number are measured differently.

Common antilog bases and when they are used

The table below summarizes the bases students most often meet. The formula is always \(b^y\), but the interpretation changes with the base.

Base 10 is often the easiest to estimate because it connects directly to decimal place value. Base \(\mathrm{e}\) is often the most important in calculus because derivatives and integrals involving \(\mathrm{e}^x\) have especially clean forms. Base 2 is central to computing because binary systems are built from two states. Custom bases are useful for algebra practice because they force you to understand the structure rather than memorize a single calculator button.

In many calculator interfaces, the buttons are labeled differently. A button written as \(10^x\) gives the common antilog. A button written as \(\mathrm{e}^x\) gives the natural antilog. A button written as \(y^x\), \(x^y\), or a general power key can evaluate any base if you enter the base and exponent correctly. This page's calculator combines those options in one place so you can choose the base explicitly.

If you are studying the functions behind these calculations, the key relationship is that exponential and logarithmic functions are inverse functions. A logarithm answers, "what exponent gives this number?" An antilog answers, "what number comes from this exponent?" The direction changes, but the base stays the same.

Antilog versus exponentiation: what is the difference?

Every antilog is an exponentiation, but not every exponentiation is described as an antilog. The word antilog is used when the exponent came from a logarithm and you are reversing that logarithm. If someone simply asks for \(3^4\), you are evaluating a power. If someone says \(\log_3(x)=4\) and asks for \(x\), then \(x=3^4\) is an antilog step because it undoes the logarithm.

This distinction helps with search intent and with mathematical communication. A general exponent problem may focus on rules such as \(a^m a^n=a^{m+n}\), fractional powers, or negative exponents. An antilog problem focuses on reversing a log-scale value. The calculation overlaps, but the reason for doing it is different.

For example, a finance problem might model growth using a logarithm because the log transformation makes percentage changes easier to compare. A chemistry problem might use pH because hydrogen ion concentration spans many powers of ten. A sound problem might use decibels because intensity ratios can be extremely large. In all of these cases, the antilog is the bridge back from the compact logarithmic scale to the original quantity.

When writing a solution, use language that matches the problem. If you are solving a logarithmic equation, say "take the antilog" or "rewrite in exponential form." If you are simplifying powers, say "use exponent rules." Clear wording prevents confusion between the operation itself and the reason the operation is needed.

How to check an antilog answer without starting over

There are three quick checks that work for most antilog answers. First, check the sign and size. For a base greater than 1, a positive logarithm should give a result greater than 1, and a negative logarithm should give a result between 0 and 1. Second, check the interval. If the base is 10 and \(y\) is between 3 and 4, then the answer must be between 1000 and 10000. Third, apply the logarithm again. If \(x=b^y\), then \(\log_b(x)\) should return \(y\).

Here is a full check. Suppose you calculate \(\operatorname{antilog}_{10}(2.25)\approx177.827941\). The log value 2.25 is between 2 and 3, so the answer should be between 100 and 1000. It is. Now apply the common logarithm:

The check returns the starting value, so the result is consistent. If the answer had been 17.782794 or 1778.27941, the interval check would have caught the decimal-place error before any detailed recalculation.

For natural antilogs, use the same idea with \(\ln\). If \(x=\mathrm{e}^{3.2}\), then \(\ln(x)\) should be 3.2. For base 2, if \(x=2^9\), then \(\log_2(x)\) should be 9. The exact base changes, but the inverse relationship stays the same.

How to use an antilog calculator correctly

The most important step is choosing the same base as the logarithm. If a value was produced by a base 10 logarithm, use base 10 for the antilog. If it was produced by a natural logarithm, use base \(\mathrm{e}\). If it was produced by a binary logarithm, use base 2. A correct log value with the wrong base gives a wrong original number.

Follow this workflow:

1. Identify the logarithm value \(y\).
2. Identify the base \(b\). If the expression is \(\log_{10}(x)\), use \(b=10\). If it is \(\ln(x)\), use \(b=\mathrm{e}\).
3. Calculate \(b^y\).
4. Check whether the result is reasonable. Positive log values usually produce results greater than 1 when \(b>1\). Negative log values usually produce results between 0 and 1 when \(b>1\).
5. Substitute back using \(\log_b(x)\) if you need to verify the answer.

For example, if \(y=-3\) and \(b=10\), then \(x=10^{-3}=0.001\). A negative common logarithm corresponds to a small positive number. This is common in chemistry, probability, scientific notation, and measurements involving very small quantities.

When results are very large or very small, scientific notation may be clearer than a long decimal. For example, \(10^{-8}=0.00000001\), which is more readable as \(1\times10^{-8}\). If you need broader calculator support beyond inverse logarithms, the easy scientific calculator and math calculator are useful general tools.

Antilog and inverse functions

Antilogarithms are a direct example of inverse functions. A function and its inverse undo each other. The logarithm function \(\log_b(x)\) and the exponential function \(b^x\) are inverses when \(b>0\) and \(b\ne1\).

These identities are true for valid inputs. The logarithm \(\log_b(x)\) is defined only for \(x>0\), so an antilog result is always positive when the base is valid. This is why the output of \(b^y\) cannot be zero or negative for real \(y\) and positive base \(b\).

For a broader algebra foundation, review inverse functions and function concepts. The antilog is one specific inverse-function example, and understanding it helps with exponentials, logarithms, and equation solving.

Graphically, \(y=\log_b(x)\) and \(y=b^x\) are reflections across the line \(y=x\). This reflection is a hallmark of inverse functions. If \(b>1\), the exponential function increases quickly, while the logarithm increases slowly. The antilog calculation moves from the logarithm's output back to the exponential graph's output.

Real-world uses of antilogs

Antilogs are used whenever a logarithmic value must be converted back into the original scale. Logarithms are often used because they compress huge ranges of values, turn multiplication into addition, or make exponential relationships easier to analyze. The antilog reverses that compression.

Chemistry and pH

The pH scale is logarithmic. It is defined by:

To find hydrogen ion concentration from pH, use an antilog:

If \(\mathrm{pH}=3\), then \([H^+]=10^{-3}=0.001\). If \(\mathrm{pH}=7\), then \([H^+]=10^{-7}\). A one-unit pH change represents a factor of 10 change in hydrogen ion concentration.

Sound intensity and decibels

Decibels use logarithms to describe ratios. When a value has been converted into decibels, an antilog is needed to recover the intensity ratio. A simplified relationship is:

Solving for the ratio gives:

This is an antilog calculation because the logarithmic decibel value is being returned to a ratio.

Growth and decay models

Exponential growth and decay often involve logarithms when solving for time or rate. After taking logs to solve an equation, an antilog may be used to return to the original scale. For example, if \(\ln(A)=k\), then \(A=\mathrm{e}^k\). This appears in finance, population modeling, radioactive decay, and continuous compounding.

Statistics and transformed data

Data analysts sometimes take logarithms to reduce skewness or linearize a relationship. After modeling on the log scale, they may use an antilog to convert predictions back to the original units. The interpretation must be careful because averaging and transforming can interact in subtle ways, but the basic reverse operation is still exponentiation.

Antilog rules and useful identities

Because an antilog is an exponential expression, exponent rules apply. These rules help simplify and estimate antilog values.

These rules connect antilogs to the broader study of exponents. For more practice, see exponents and logarithms, exponents and logs, and exponent-log functions. The antilog calculator is useful for numerical answers, but the rules explain why the results behave the way they do.

Common mistakes with antilogs

Mistake 1: Using the wrong base

If the original logarithm used base 10, the antilog must use base 10. If the original logarithm used base \(\mathrm{e}\), the antilog must use \(\mathrm{e}\). Mixing bases is the most common error.

Mistake 2: Treating antilog as multiplication

The antilog is not \(b\times y\). It is \(b^y\). For example, the antilog base 10 of 3 is \(10^3=1000\), not \(10\times3=30\).

Mistake 3: Forgetting that log inputs are positive

For real logarithms, the original number \(x\) must be positive. The antilog result \(b^y\) is always positive when \(b>0\) and \(b\ne1\).

Mistake 4: Confusing \(\log\) and \(\ln\)

\(\log\) often means base 10, while \(\ln\) means base \(\mathrm{e}\). Their antilogs are \(10^y\) and \(\mathrm{e}^y\), respectively.

Mistake 5: Rounding too early

Antilog values can be sensitive to decimal places. Keep more digits during intermediate steps and round only at the end unless the problem instructs otherwise.

Mistake 6: Allowing base 1

Base 1 is not a valid logarithm base. Since \(1^y=1\) for every \(y\), it cannot produce a one-to-one inverse relationship.

Antilog in equation solving

Antilogs are often used when solving logarithmic equations. The goal is to isolate the logarithm, then rewrite the equation in exponential form. This is the same as taking the antilog of both sides with the correct base.

Example: solve \(\log_{10}(x)=2.8\)

The solution is positive, which matches the domain of the logarithm.

Example: solve \(\ln(x-3)=1.5\)

After using the antilog, remember to solve for \(x\), not just the expression inside the logarithm.

Example: solve \(\log_2(3x)=5\)

For more equation practice, review exponential and logarithmic equations and the twelfth-grade version of exponential and logarithmic equations.

Practice questions

1. Find \(\operatorname{antilog}_{10}(3)\).
2. Find \(\operatorname{antilog}_{10}(-2)\).
3. Find \(x\) if \(\log_{10}(x)=4.2\).
4. Find \(x\) if \(\ln(x)=3\).
5. Find \(x\) if \(\log_2(x)=9\).
6. Find \(x\) if \(\log_5(x)=2.5\).
7. Explain why base 1 is not allowed for logarithms or antilogs.
8. Convert \(\mathrm{pH}=5\) into hydrogen ion concentration using \([H^+]=10^{-\mathrm{pH}}\).
9. Solve \(\ln(x+2)=2\).
10. Use a reasonableness check: should \(10^{1.3}\) be between 10 and 100? Explain.

Answer checks

1. \(10^3=1000\).
2. \(10^{-2}=0.01\).
3. \(x=10^{4.2}\approx15848.931925\).
4. \(x=\mathrm{e}^3\approx20.085537\).
5. \(x=2^9=512\).
6. \(x=5^{2.5}\approx55.901699\).
7. Because \(1^y=1\) for every \(y\), so the inverse would not be unique.
8. \([H^+]=10^{-5}\).
9. \(x+2=\mathrm{e}^2\), so \(x=\mathrm{e}^2-2\).
10. Yes. \(1.3\) is between 1 and 2, so \(10^{1.3}\) is between \(10^1\) and \(10^2\).

Quick quiz

Question: If \(\log_{10}(x)=2.5\), what expression gives \(x\)?

Questions and answers

What is the antilog formula?

The antilog formula is \(\operatorname{antilog}_b(y)=b^y\). If \(\log_b(x)=y\), then \(x=b^y\).

What is the antilog of 2 in base 10?

The antilog base 10 of 2 is \(10^2=100\).

What is the natural antilog?

The natural antilog is \(\mathrm{e}^y\), the inverse of \(\ln(x)\).

What is the difference between log and antilog?

A log converts a number into an exponent. An antilog converts the exponent back into the original number using the same base.

Can an antilog be negative?

For real logarithms with a valid positive base, an antilog result is always positive because \(b^y>0\).

How do I calculate antilog on a calculator?

Use the power function. For base 10, calculate \(10^y\). For natural antilog, calculate \(\mathrm{e}^y\). For base 2, calculate \(2^y\).

Is antilog the same as inverse logarithm?

Yes. Antilogarithm and inverse logarithm describe the same reverse operation: \(b^y\), where \(y\) is the logarithm value.

What base should I use?

Use the base of the original logarithm. Use 10 for common logs, \(\mathrm{e}\) for natural logs, and 2 for binary logs.

Key takeaways

• The antilog is the inverse of a logarithm.
• If \(\log_b(x)=y\), then \(x=b^y\).
• Base 10 antilog is \(10^y\).
• Natural antilog is \(\mathrm{e}^y\).
• Base 2 antilog is \(2^y\).
• The base must be positive and cannot equal 1.
• Antilog outputs are positive for real inputs and valid positive bases.
• Use antilogs to solve logarithmic equations and convert logarithmic-scale values back to original units.

Continue building the topic by reviewing logarithms, logarithmic functions, and exponential functions. These pages explain the forward logarithm and exponential models that make antilog calculations meaningful.