An Expression Upstairs

The exponent can be a whole expression. In ${2}^{3x+1} = 16$, all of $3x+1$ sits upstairs. The method does not change: write 16 as ${2}^{4}$, so the equation reads ${2}^{3x+1} = {2}^{4}$.

Equating the exponents now gives an ordinary linear equation, $3x+1 = 4$. Move the 1, then divide by 3: $3x = 3$, so x = 1. Check it in the original: ${2}^{3+1}$ is ${2}^{4}$, which is 16.

The whole expression is the exponent, not just its first term. That is why it is written in brackets after the caret. Equating exponents hands the whole bracket over to an equation you already know how to solve.