Question

Write each expression as a sum and/or difference of logarithms. Express powers as factors.

${\log }_3\frac{x}{9}$

Short answer

The result is${\log }_{3}x+2$.

Step-by-step solution

Step 1. Given information.

Consider the given information.

${\log }_3\frac{x}{9}$

Step 2. Calculation.

Apply the logarithm property to simplify.

$\begin{array}{l}{\log }_3\frac{x}{9}={\log }_{3}x+{\log }_{3}9\\ ={\log }_{3}x+{\log }_3{3}^2\\ ={\log }_{3}x+2{\log }_{3}3\\ ={\log }_{3}x+2\end{array}$

Step 3. Conclusion.

The obtained result is${\log }_{3}x+2$.