Properties of Logarithms

Definition

$y = \log_a(x) \quad \text{means} \quad a^y = x$

where $a > 0$, $a \neq 1$, and $x > 0$

Basic Properties

Property	Formula
Log of 1	$\log_a(1) = 0$
Log of base	$\log_a(a) = 1$
Inverse	$a^{\log_a(x)} = x$
Inverse	$\log_a(a^x) = x$

Logarithm Rules

Product Rule

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

Quotient Rule

$\log_a\left(\frac{x}{y}\right) = \log_a(x) - \log_a(y)$

Power Rule

$\log_a(x^n) = n \cdot \log_a(x)$

Root Rule

$\log_a(\sqrt[n]{x}) = \frac{\log_a(x)}{n}$

Change of Base Formula

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

Common Logarithms

Notation	Base
$\log(x)$	Base 10 (common log)
$\ln(x)$	Base $e$ (natural log)
$\log_2(x)$	Base 2 (binary log)

Natural Logarithm Properties

Property	Value
$\ln(1)$	$0$
$\ln(e)$	$1$
$\ln(e^x)$	$x$
$e^{\ln x}$	$x$
$e$	$\approx 2.71828$

Solving Logarithmic Equations

Basic Form

$\log_a(x) = c$

Solution: $x = a^c$

Using Properties

$\log_a(x) + \log_a(y) = c$

$\log_a(xy) = c$

$xy = a^c$

Solving Exponential Equations

Take the logarithm of both sides:

$a^x = b$

$x \cdot \log(a) = \log(b)$

$x = \frac{\log(b)}{\log(a)}$

Common Log Values

$x$	$\log_{10}(x)$
$1$	$0$
$10$	$1$
$100$	$2$
$1000$	$3$
$0.1$	$-1$
$0.01$	$-2$

Domain Restrictions

For $\log_a(f(x))$:

• $f(x)$ must be positive: $f(x) > 0$
• The base $a$ must be positive and not 1