When a Logarithm Is the Exponent — Cheat sheet

The three

a^{\log_a x} = x

a^{m \log_a x} = x^m

a^{\log_b c} = c^{\log_b a}

The first needs matching bases. The second is the first with an outside power. The third trades the two ends when nothing matches.

8^{\log_2 5} = 2^{3 \log_2 5} = \left(2^{\log_2 5}\right)^3 = 5^3 = 125

Split the sum first, then cancel.

2^{\log_2 5 + 3} = 2^{\log_2 5} \cdot 2^3 = 40

5^{\log_5 2 - 1} = \frac{2}{5}

a^{\log_b c} = c^{\log_b a}

\text{so}

81^{\log_3 2} = 2^{\log_3 81} = 2^4 = 16

$2^{\log_3 7}$ cancels to nothing. The bases share no common power here.
A negative exponent is a reciprocal, not a negative answer: $3^{-2 \log_3 2} = 2^{-2} = \dfrac{1}{4}$ .
$2^{\log_2 5 + 3}$ is $40$ , not $8$ . The $3$ multiplies, it does not vanish.
$a^{m \log_a x} = x^m$ , never $m x$ .