Chapter 5, Problem 8CT

In Problems $8-11$, evaluate each expression, without using a calculator.

${\log }_6\frac{1}{36}$

To determine

To calculate: The value of the expression ${\log }_6\frac{1}{36}$.

Answer to Problem 8CT

Solution:

${\log }_6\frac{1}{36}=-2$.

Explanation of Solution

Given information:

Given expression: ${\log }_6\frac{1}{36}$.

Formula used:

Negative exponent rule:

$a^{-u}=\frac{1}{a^u}$

Logarithmic properties:

${\log }_a{M}^r=r{\log }_{a}M$

${\log }_{a}a=1$

Calculation:

Use negative exponent rule for the fraction $\frac{1}{36}$,

$\frac{1}{36}={36}^{-1}$

From this, the expression becomes:

$\Rightarrow {\log }_6{36}^{-1}$

$\Rightarrow {\log }_6{\left(6^2\right)}^{-1}$

$\Rightarrow {\log }_6\left(6^{-2}\right)$

Use the properties of logarithm, apply ${\log }_a{M}^r=r{\log }_{a}M$,

$\Rightarrow {\log }_6\left(6^{-2}\right)=-2{\log }_{6}6$

Now, apply logarithmic property ${\log }_{a}a=1$,

$\Rightarrow -2{\log }_{6}6=-2\cdot 1=-2$

Therefore, the value of expression ${\log }_6\frac{1}{36}$ is $-2$.