Evaluate 1/2 log base 5 of 15- log base 5 of square root of 75

Simplify ${\displaystyle \frac{1}{2}{\log }_5\left(15\right)}$ by moving ${\displaystyle \frac{1}{2}}$ inside the logarithm.

Rewrite ${\displaystyle 75}$ as ${\displaystyle 5^2\cdot 3}$.

Pull terms out from under the radical.

Multiply ${\displaystyle \frac{{15}^{\frac{1}{2}}}{5\sqrt{3}}}$ by ${\displaystyle \frac{\sqrt{3}}{\sqrt{3}}}$.

Move ${\displaystyle \sqrt{3}}$.

Raise ${\displaystyle \sqrt{3}}$ to the power of ${\displaystyle 1}$.

Raise ${\displaystyle \sqrt{3}}$ to the power of ${\displaystyle 1}$.

Use the power rule ${\displaystyle a^m{a}^n=a^{m+n}}$ to combine exponents.

Add ${\displaystyle 1}$ and ${\displaystyle 1}$.

Rewrite ${\displaystyle {\sqrt{3}}^2}$ as ${\displaystyle 3}$.

Multiply ${\displaystyle 5}$ by ${\displaystyle 3}$.

Move ${\displaystyle {15}^{\frac{1}{2}}}$ to the denominator using the negative exponent rule ${\displaystyle b^n=\frac{1}{b^{-n}}}$.

Multiply ${\displaystyle 15}$ by ${\displaystyle {15}^{-\frac{1}{2}}}$.

Write ${\displaystyle 1}$ as a fraction with a common denominator.

Combine the numerators over the common denominator.

Subtract ${\displaystyle 1}$ from ${\displaystyle 2}$.

The change of base rule can be used if ${\displaystyle a}$ and ${\displaystyle b}$ are greater than ${\displaystyle 0}$ and not equal to ${\displaystyle 1}$, and ${\displaystyle x}$ is greater than ${\displaystyle 0}$.

Substitute in values for the variables in the change of base formula, using ${\displaystyle b=10}$.