Question

In the following exercises, find the exact value of each logarithm without using a calculator.

${\log }_3\frac{1}{27}$

Short answer

The value of expression${\log }_3\frac{1}{27}is-3.$

Step-by-step solution

Step 1. Given

An expression is given as

${\log }_3\frac{1}{27}$

Step 2. Calculation

Put the expression equal to $x$,

${\log }_3\frac{1}{27}$

Change the equation into exponential form as,

$$\begin{gathered} 3^x=\frac{1}{27} \\ {3}^x=\frac{1}{{\left(3\right)}^3} \\ {3}^x=3^{-1} \\ x=-1 \end{gathered}$$