Math ↠ Logarithms ↠ Logarithm

Logarithm

If you use logarithm, the exponent is determined using the base and the potency value.

$b^\color{red}{x}=a\Leftrightarrow \color{red}{x} = \log_b(a)$
(say: Logarithm of a to base b)

$x ...$ the exponent
$b ...$ the base
$a ...$ the potency value

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Remember

For $\log_b{a}=x$ the following applies:

$a, b > 0$ and $b\neq1$

Example: $2^3=8$ and the logarithm of 8 to base 2 is: $\log_2 8=3$.

The brackets in logarithms can be left out if there are no misunderstandings e.g. $\log_2 8$ instead of $\log_2 (8)$

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Hint

  • If you use exponentiation, you search the potency value:
    $b^n=\color{red}{x}$
  • If you useroot extraction, you search the base:
    $\color{red}{x}^n=a$
  • If you use logarithms, you search the exponent:
    $b^\color{red}{x}=a$

Example

  • $\log_3(81)=4$, because $3^4=81$
  • $\log_5(-25)=\text{not defiened}$
  • $\log_4(\frac{1}{16})=-2$, because $4^{-2}=\frac{1}{16}$