Defining the Logarithm — Cheat sheet

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Definition

$\log_a x = y \iff a^y = x$ , with $a > 0$ , $a \ne 1$ and $x > 0$ .

Out loud: $\log_a x$ is the power that turns $a$ into $x$ .

The base stays the base. Only the other two change places.

$\log_a 1 = 0$ and $\log_a a = 1$ , for every allowed base.

Part	Condition	Why
base $a$	$a > 0$ , $a \ne 1$	as for the exponential it inverts
argument $x$	$x > 0$	no power of a positive base is $0$ or negative
answer $y$	anything	a logarithm may be negative or fractional

$\log_a(a^x) = x$ and $a^{\log_a x} = x$ . The graphs are mirror images in $y = x$ .