Question

Evaluate each expression.

${\log }_{9}3$

Short answer

The value of ${\log }_{9}3=\frac{1}{2}$.

Step-by-step solution

Step 1. Given Information.

Given, to evaluate the expression ${\log }_{9}3$.

Step 2. Calculation.

According to the properties of logarithms, ${\log }_b{a}^n=n{\log }_{b}a,{\log }_{a}a=1$

Simplifying the expression:

$\begin{array}{l}{\log }_{9}3={\log }_9\sqrt{9}\\ {\log }_{9}3={\log }_9{\left(9\right)}^{\frac{1}{2}}\\ {\log }_{9}3=\frac{1}{2}{\log }_9\left(9\right)\\ {\log }_{9}3=\frac{1}{2}\left(1\right)\\ {\log }_{9}3=\frac{1}{2}\end{array}$

Step 3. Conclusion.

Hence, ${\log }_{9}3=\frac{1}{2}$.