Log Calculator: Solve Logarithmic Equations Step by Step
A log calculator is a mathematical tool that computes logarithms (the inverse operation of exponentiation), solving equations of the form \( \log_b(x) = y \) where \( b^y = x \), including common logarithms (base 10), natural logarithms (base e), and logarithms of any base. This calculator evaluates logarithmic expressions, solves logarithmic equations using properties like product rule, quotient rule, and power rule, applies change of base formula, and provides step-by-step solutions for algebra students, calculus courses, exponential growth problems, pH calculations, decibel measurements, computer science applications, and any mathematical context requiring logarithmic computation, equation solving, or transformation between exponential and logarithmic forms.
📊 Interactive Logarithm Calculator
Calculate logarithms and solve equations with detailed steps
Basic Logarithm Calculator
Calculate: \( \log_b(x) \)
Natural Logarithm (ln) Calculator
Calculate: \( \ln(x) = \log_e(x) \) where \( e \approx 2.71828 \)
Change of Base Formula
Convert: \( \log_b(x) = \frac{\log_c(x)}{\log_c(b)} \)
Antilog Calculator (Inverse Log)
If \( \log_b(x) = y \), then antilog: \( x = b^y \)
Solve Logarithmic Equation
Solve: \( \log_b(x) = y \) for x
Understanding Logarithms
A logarithm is the inverse operation of exponentiation. If \( b^y = x \), then \( \log_b(x) = y \). The logarithm answers the question: "To what power must we raise the base to get this number?"
Logarithm Definition and Notation
Logarithm Definition:
\[ \log_b(x) = y \iff b^y = x \]
Where:
\( b \) = base (must be positive, \( b \neq 1 \))
\( x \) = argument (must be positive)
\( y \) = result (can be any real number)
Read as: "log base b of x equals y"
Types of Logarithms
Common Logarithm (Base 10)
Common Logarithm: \( \log(x) = \log_{10}(x) \)
When no base is written, base 10 is assumed
Examples:
\( \log(100) = 2 \) because \( 10^2 = 100 \)
\( \log(1000) = 3 \) because \( 10^3 = 1000 \)
Natural Logarithm (Base e)
Natural Logarithm: \( \ln(x) = \log_e(x) \)
where \( e \approx 2.71828... \) (Euler's number)
Examples:
\( \ln(e) = 1 \) because \( e^1 = e \)
\( \ln(e^2) = 2 \) because \( e^2 = e^2 \)
Binary Logarithm (Base 2)
Binary Logarithm: \( \log_2(x) \)
Common in computer science
Examples:
\( \log_2(8) = 3 \) because \( 2^3 = 8 \)
\( \log_2(64) = 6 \) because \( 2^6 = 64 \)
Logarithm Properties and Rules
Fundamental Properties
| Property | Formula | Example |
|---|---|---|
| Product Rule | \( \log_b(xy) = \log_b(x) + \log_b(y) \) | \( \log(6) = \log(2) + \log(3) \) |
| Quotient Rule | \( \log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y) \) | \( \log(5) = \log(10) - \log(2) \) |
| Power Rule | \( \log_b(x^n) = n \cdot \log_b(x) \) | \( \log(100) = 2\log(10) = 2 \) |
| Change of Base | \( \log_b(x) = \frac{\log_c(x)}{\log_c(b)} \) | \( \log_2(8) = \frac{\log(8)}{\log(2)} \) |
| Log of 1 | \( \log_b(1) = 0 \) | \( \log(1) = 0 \) |
| Log of Base | \( \log_b(b) = 1 \) | \( \log(10) = 1 \) |
| Inverse Property | \( b^{\log_b(x)} = x \) | \( 10^{\log(5)} = 5 \) |
Step-by-Step Examples
Example 1: Basic Logarithm
Problem: Calculate \( \log_2(16) \)
Question: To what power must we raise 2 to get 16?
Step 1: Try powers of 2
\( 2^1 = 2, \quad 2^2 = 4, \quad 2^3 = 8, \quad 2^4 = 16 \)
Step 2: Identify the power
Since \( 2^4 = 16 \), we have \( \log_2(16) = 4 \)
Answer: \( \log_2(16) = 4 \)
Example 2: Using Product Rule
Problem: Simplify \( \log(20) \) using \( \log(2) \approx 0.301 \) and \( \log(5) \approx 0.699 \)
Step 1: Factor 20
\( 20 = 2 \times 10 = 2 \times 2 \times 5 = 4 \times 5 \)
Step 2: Apply product rule
\( \log(20) = \log(4 \times 5) = \log(4) + \log(5) \)
Step 3: Simplify \( \log(4) \)
\( \log(4) = \log(2^2) = 2\log(2) = 2(0.301) = 0.602 \)
Step 4: Add values
\( \log(20) = 0.602 + 0.699 = 1.301 \)
Answer: \( \log(20) \approx 1.301 \)
Example 3: Solving Logarithmic Equation
Problem: Solve \( \log_3(x) = 4 \)
Step 1: Convert to exponential form
If \( \log_3(x) = 4 \), then \( 3^4 = x \)
Step 2: Calculate \( 3^4 \)
\( 3^4 = 3 \times 3 \times 3 \times 3 = 81 \)
Answer: \( x = 81 \)
Verification: \( \log_3(81) = 4 \) ✓
Change of Base Formula
Change of Base Formula:
\[ \log_b(x) = \frac{\log_c(x)}{\log_c(b)} \]
Most commonly used with base 10 or e:
\[ \log_b(x) = \frac{\log(x)}{\log(b)} \quad \text{or} \quad \log_b(x) = \frac{\ln(x)}{\ln(b)} \]
Change of Base Example
Problem: Calculate \( \log_5(30) \) using common logarithms
Solution: Use change of base formula
\( \log_5(30) = \frac{\log(30)}{\log(5)} \)
\( \log_5(30) = \frac{1.477}{0.699} \approx 2.113 \)
Answer: \( \log_5(30) \approx 2.113 \)
Common Logarithm Values
| Number (x) | \( \log(x) \) | Number (x) | \( \log(x) \) |
|---|---|---|---|
| 1 | 0 | 100 | 2 |
| 2 | 0.301 | 1,000 | 3 |
| 3 | 0.477 | 10,000 | 4 |
| 5 | 0.699 | 100,000 | 5 |
| 10 | 1 | 1,000,000 | 6 |
Natural Logarithm Values
| Number (x) | \( \ln(x) \) | Number (x) | \( \ln(x) \) |
|---|---|---|---|
| 1 | 0 | 10 | 2.303 |
| 2 | 0.693 | 20 | 2.996 |
| e (2.718...) | 1 | 50 | 3.912 |
| 5 | 1.609 | 100 | 4.605 |
Logarithm and Exponential Relationship
Logarithms are Inverse of Exponentials:
\[ y = b^x \iff x = \log_b(y) \]
Inverse Properties:
\( \log_b(b^x) = x \) for all \( x \)
\( b^{\log_b(x)} = x \) for \( x > 0 \)
Real-World Applications
Science and Engineering
- pH scale: \( pH = -\log[H^+] \) (hydrogen ion concentration)
- Richter scale: Earthquake magnitude (logarithmic)
- Decibels: \( dB = 10\log\left(\frac{I}{I_0}\right) \) (sound intensity)
- Radioactive decay: Half-life calculations
- Population growth: Exponential and logarithmic models
Mathematics and Computer Science
- Algorithm complexity: O(log n) time complexity
- Information theory: Entropy and data compression
- Number theory: Prime number distribution
- Calculus: Integration and differentiation
Finance and Economics
- Compound interest: Time to double investment
- Present value: Discounting future cash flows
- Log returns: Financial market analysis
Common Mistakes to Avoid
⚠️ Frequent Errors
- Log of negative: \( \log(x) \) undefined for \( x \leq 0 \)
- Log of zero: \( \log(0) \) is undefined
- Base restrictions: Base must be positive and ≠ 1
- Product confusion: \( \log(x + y) \neq \log(x) + \log(y) \)
- Power confusion: \( \log(x^2) = 2\log(x) \), not \( (\log x)^2 \)
- Base mixing: Can't add logs with different bases directly
- Notation: \( \log(x) \) usually means base 10, not base e
Tips for Working with Logarithms
Best Practices:
- Memorize key values: log(1)=0, log(10)=1, ln(e)=1
- Use properties: Simplify before calculating
- Change of base: Convert to log or ln for calculation
- Check domain: Ensure arguments are positive
- Convert equations: Use exponential form to solve
- Verify answers: Substitute back into original equation
- Understand inverse: Log undoes exponentiation
Solving Logarithmic Equations
Method 1: Convert to Exponential Form
Solve: \( \log_2(x) = 5 \)
Step 1: Convert: \( 2^5 = x \)
Step 2: Calculate: \( x = 32 \)
Method 2: Use Logarithm Properties
Solve: \( \log(x) + \log(x-3) = 1 \)
Step 1: Product rule: \( \log[x(x-3)] = 1 \)
Step 2: Exponential: \( x(x-3) = 10^1 = 10 \)
Step 3: Expand: \( x^2 - 3x = 10 \)
Step 4: Rearrange: \( x^2 - 3x - 10 = 0 \)
Step 5: Factor: \( (x-5)(x+2) = 0 \)
Step 6: Solutions: \( x = 5 \) or \( x = -2 \)
Check domain: \( x = -2 \) invalid (negative), so \( x = 5 \)
Frequently Asked Questions
What is a logarithm and how does it work?
A logarithm is the inverse of exponentiation. log_b(x) = y means "b raised to what power equals x?" Answer is y. Example: log₁₀(100) = 2 because 10² = 100. Logarithms convert multiplication into addition, making complex calculations easier. Used extensively in science, engineering, and mathematics for exponential relationships, growth/decay, and scale transformations like pH and decibels.
What is the difference between log and ln?
log (common logarithm) uses base 10: log(x) = log₁₀(x). ln (natural logarithm) uses base e (≈2.718): ln(x) = logₑ(x). Both follow same rules but different bases. log common in engineering/science; ln used in calculus, continuous growth. Convert: ln(x) = log(x)/log(e) ≈ 2.303·log(x). Calculator buttons: LOG for base 10, LN for base e.
How do you solve logarithmic equations?
Method 1: Convert to exponential form. If log_b(x) = y, then x = b^y. Method 2: Use log properties to combine/simplify, then convert. Method 3: Take log of both sides. Always check solutions in original equation - reject negatives/zeros (undefined for log). Example: log₂(x) = 3 → x = 2³ = 8. Verify domain restrictions before finalizing answer.
What is the change of base formula?
Change of base formula converts any logarithm to different base: log_b(x) = log_c(x)/log_c(b). Most commonly convert to base 10 or e for calculator use: log_b(x) = log(x)/log(b) or ln(x)/ln(b). Example: log₅(30) = log(30)/log(5) = 1.477/0.699 ≈ 2.113. Essential when calculator lacks arbitrary base function. Works for any valid base.
What are the main logarithm properties?
Product rule: log(xy) = log(x) + log(y). Quotient rule: log(x/y) = log(x) - log(y). Power rule: log(x^n) = n·log(x). Special values: log(1) = 0, log(b) = 1 for log base b. Inverse: b^(log_b(x)) = x. These properties allow algebraic manipulation and simplification of logarithmic expressions. Memorize for efficient problem solving.
Why can't you take the log of a negative number?
Logarithms defined only for positive numbers because b^y (for positive base b) is always positive for any real y. No real power of positive number gives negative result. Example: 10^x can never equal -100, so log(-100) undefined. In complex numbers, log extends to negatives using imaginary numbers, but standard real logarithms require positive arguments. Domain restriction: x > 0.
Key Takeaways
Logarithms are the inverse of exponentiation, answering "to what power must the base be raised to get this number?" Understanding logarithm properties, rules, and applications enables solving exponential equations and working with exponential relationships across mathematics and science.
Essential principles to remember:
- Definition: \( \log_b(x) = y \iff b^y = x \)
- Common log: \( \log(x) = \log_{10}(x) \)
- Natural log: \( \ln(x) = \log_e(x) \)
- Product rule: \( \log(xy) = \log(x) + \log(y) \)
- Quotient rule: \( \log(x/y) = \log(x) - \log(y) \)
- Power rule: \( \log(x^n) = n\log(x) \)
- Change of base: \( \log_b(x) = \frac{\log(x)}{\log(b)} \)
- Domain: x must be positive for log(x)
- Inverse property: \( b^{\log_b(x)} = x \)
- Special values: \( \log(1) = 0, \log(10) = 1, \ln(e) = 1 \)
Getting Started: Use the interactive log calculator at the top of this page to compute logarithms of any base, natural logarithms, apply change of base formula, calculate antilog, and solve logarithmic equations. Enter your values and receive instant results with step-by-step explanations showing the complete solution process.
