Exponents & Logarithms Formulas
IB Mathematics Analysis & Approaches
⚡ Laws of Exponents
Product Rule:
\[a^x \cdot a^y = a^{x+y}\]
Multiply powers with the same base by adding the exponents
Quotient Rule:
\[\frac{a^x}{a^y} = a^{x-y}\]
Divide powers with the same base by subtracting the exponents (where \(a \neq 0\))
Power Rule:
\[(a^x)^y = a^{xy}\]
Raise a power to another power by multiplying the exponents
Power of a Product:
\[(ab)^x = a^x \cdot b^x\]
Distribute the exponent to each factor inside the parentheses
Power of a Quotient:
\[\left(\frac{a}{b}\right)^x = \frac{a^x}{b^x}\]
Distribute the exponent to both numerator and denominator (where \(b \neq 0\))
Zero Exponent:
\[a^0 = 1\]
Any nonzero base raised to the power of zero equals 1 (where \(a \neq 0\))
Negative Exponent:
\[a^{-x} = \frac{1}{a^x}\]
A negative exponent indicates the reciprocal of the positive exponent (where \(a \neq 0\))
Fractional Exponent:
\[a^{\frac{m}{n}} = \sqrt[n]{a^m} = \left(\sqrt[n]{a}\right)^m\]
A fractional exponent represents a root; the denominator is the root index
📘 Introduction to Logarithms
Definition:
A logarithm is the inverse operation of exponentiation. It answers: "To what power must the base be raised to get a given number?"
\[a^x = b \iff \log_a(b) = x\]
where \(a > 0\), \(a \neq 1\), and \(b > 0\)
Common Logarithm Bases:
Base 10 (Common Logarithm): \(\log_{10}(x)\) written as \(\log(x)\)
Base \(e\) (Natural Logarithm): \(\log_e(x)\) written as \(\ln(x)\), where \(e \approx 2.718\)
📐 Laws of Logarithms
Product Rule:
\[\log_a(xy) = \log_a(x) + \log_a(y)\]
The log of a product equals the sum of the logs
Quotient Rule:
\[\log_a\left(\frac{x}{y}\right) = \log_a(x) - \log_a(y)\]
The log of a quotient equals the difference of the logs
Power Rule:
\[\log_a(x^m) = m \cdot \log_a(x)\]
The log of a power equals the exponent times the log of the base
Change of Base Formula:
\[\log_a(x) = \frac{\log_c(x)}{\log_c(a)}\]
Convert logarithms to any base \(c\) (commonly base 10 or \(e\))
⭐ Special Properties
Logarithm of 1:
\[\log_a(1) = 0\]
For any base \(a > 0\), \(a \neq 1\)
Logarithm of the Base:
\[\log_a(a) = 1\]
For any base \(a > 0\), \(a \neq 1\)
Inverse Property (Exponent-Log):
\[a^{\log_a(x)} = x\]
where \(x > 0\)
Inverse Property (Log-Exponent):
\[\log_a(a^x) = x\]
For any real number \(x\)
🌿 Natural Logarithm Properties
Natural Log of \(e\):
\[\ln(e) = 1\]
Natural Log of 1:
\[\ln(1) = 0\]
Inverse Property:
\[e^{\ln(x)} = x\]
\[\ln(e^x) = x\]
Converting to Natural Log:
\[\log_a(x) = \frac{\ln(x)}{\ln(a)}\]
🔑 Quick Reference Table
| Exponential Form | Logarithmic Form |
|---|---|
| \(a^x = b\) | \(\log_a(b) = x\) |
| \(10^x = y\) | \(\log(y) = x\) |
| \(e^x = y\) | \(\ln(y) = x\) |
| \(2^3 = 8\) | \(\log_2(8) = 3\) |
💡 Important: Always remember that logarithms are only defined for positive numbers, and the base must be positive and not equal to 1.