Simplify $\displaystyle{2^{\log_2 5x}}$

$\require{cancel}$Recall an exponential function and a logarithmic function of the same base are inverse functions and will thus "cancel each other out".

In one direction this means that $b^{\log_b x} = \cancel{b^{\log_b}} \vphantom{0}^{x} = x$.


$$2^{\log_2 5x} = \cancel{2^{\log_2}} \vphantom{0}^{5x} = 5x$$

Alternate Solution

This should also make sense if we think about $\log_b x$ being the "exponent needed on $b$ to produce $x$".

If we have $2^{\log_2 5x}$, then we have a $2$ raised to the precisely the power that is needed on $2$ to produce $5x$ -- so of course it should simplify to $5x$!