Math - Logarithm

The logarithm determines how many times a number (the base) must be raised to a power to produce another number (the argument).

Logarithms & Exponents

$\log_bx=y\\ b^y=x$

Negative Logarithm

The negative logarithm indicates that the number is less than 1.

$-\log_bx = \log_b\frac{1}{b}$

Logarithm 0 & 1

The logarithm of 1 is always 0

$\log_b1 = 0$
$b^0 = 1$

The logarithm with a base equal to the argument is always 1.

$\log_bb = 1$
$b^1 = b$

Natural Logarithm

The natural logarithm is a logarithm with base $e$ (Euler’s number).

Logarithm Rules

• Product Rule:
  $\log_nxy = \log_nx + \log_ny$
• Quotient Rule:
  $\log_n\frac{x}{y} = \log_nx - \log_ny$
• Power Rule:
  $\log_nx^y = y \times log_nx$
• Negative Rule:
  $-\log_nx = \log_n\frac{1}{x}$
• Change of Base Rule:
  $\log_nx = \frac{\log x}{\log n}$