Answer
False.
Work Step by Step
Take $b=10$ and $t=2$. $b^t=10^2=100$, so $\log(b^t)=\log(100)=2$. For the second equation, $\log (b)=\log (10)=1$ so $(\log b)^t=(1)^2=1$. These values aren't the same, so the equations cannot be the same.
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