The Inverse of a Logarithmic Function — Cheat sheet

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The move

$x$ is stuck inside a logarithm, so the exponential frees it.

y = \log_a x \iff x = a^y

$2^{3\log_2 x} \ne x$ . Divide by the $3$ before raising the base:

3\log_2 x = y \;\Rightarrow\; \log_2 x = \frac{y}{3} \;\Rightarrow\; x = 2^{y/3}

$5^{\log_5(x - 2)} = x - 2$ . The bracket arrives intact; the $-2$ comes off afterwards.

	domain	range
$f(x) = \log_a x$	$x > 0$	every real number
$f^{-1}(x) = a^x$	every real number	$y > 0$

$x$ sits...	...so use
in an exponent	the logarithm
inside a logarithm	the exponential