Answer
$\dfrac{3}{2}$
Work Step by Step
Rewrite the radical as an exponential expression:
$\log_5 {\left(5 \cdot 5^{\frac{1}{2}}\right)}$
If two exponential expressions having the same base are multiplied together, add the exponents, keeping the base as-is:
$=\log_5 {5^{\frac{3}{2}}}$
Let $y=\log_5{5^{\frac{3}{2}}}$.
Use the definition of logarithm $\log_b {x} = y \longleftrightarrow b^{y} = x$ to write an exponential equation.
In this exercise, the base $b$ is $5$, $y$ is the exponent, and $x$ is $5^{\frac{3}{2}}$:
$5^{y} = 5^{\frac{3}{2}}$
If two numbers having the same base are equal, that means that their exponents are also the same, so set the exponents equal to one another to solve for $y$:
$y = \frac{3}{2}$
Thus, $\log_5{5\sqrt5}=\log_5{\left(5^{\frac{3}{2}}\right)}=\frac{3}{2}$.
