Logarithmic Functions

A logarithm answers: "What exponent gives me this result?"

$\log_b(x) = y \iff b^y = x$

Reading Logarithms

$\log_2(8) = 3$ means "2 to what power equals 8?" Answer: 3

Special Logarithms

Common Log

$\log(x) = \log_{10}(x)$

Natural Log

$\ln(x) = \log_e(x)$

Logarithm Properties

$\log_b(1) = 0$ (because $b^0 = 1$) $\log_b(b) = 1$ (because $b^1 = b$) $\log_b(b^x) = x$ $b^{\log_b(x)} = x$

Logarithm Laws

Product Rule

$\log_b(xy) = \log_b(x) + \log_b(y)$

Quotient Rule

$\log_b(\frac{x}{y}) = \log_b(x) - \log_b(y)$

Power Rule

$\log_b(x^n) = n\log_b(x)$

Change of Base

$\log_b(x) = \frac{\log_a(x)}{\log_a(b)} = \frac{\ln(x)}{\ln(b)}$

Solving Exponential Equations

To solve $2^x = 10$:

1. Take log of both sides: $\log(2^x) = \log(10)$
2. Apply power rule: $x\log(2) = 1$
3. Solve: $x = \frac{1}{\log(2)} \approx 3.32$

Graph of Logarithm

$y = \log_b(x)$ is the reflection of $y = b^x$ over $y = x$

• Domain: $(0, \infty)$
• Range: all real numbers
• Vertical asymptote at $x = 0$
• Passes through $(1, 0)$