Logarithm

The logarithm serves as the inverse operation to exponentiation.

Essentially, it helps determine the exponent needed to produce a given number from a specified base.

The default log is 10: $\log{x} = \log_{10}{x}$

Logarithms & Exponents

The exponentiation of 4 to the power of 3 results in 64
The logarithm with base 4 of 64 gives 3.

Negative Logarithm

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

$-\log_nx = \log_n\frac{1}{n}$

Logarithm 0 & 1

The logarithm of 1 is always 0\

$\log_n1 = 0$
$n^0 = 1$

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

$\log_nn = 1$
$n^1 = n$

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:
  $\log_nx = \frac{\log x}{\log n}$