Question

use the change of base formula to compute \\(\log_{6} \frac{1}{8}\\). round your answer to the nearest thousandth.

Explanation:

Step1: Recall change of base formula

The change of base formula for logarithms is $\log_b a=\frac{\log_c a}{\log_c b}$ (where $c>0,c
eq1$). We can use base 10 or base $e$ (natural logarithm). Let's use base 10 here. So for $\log_6 \frac{1}{8}$, we have $a = \frac{1}{8}$ and $b = 6$. So it becomes $\frac{\log_{10}\frac{1}{8}}{\log_{10}6}$.

Step2: Calculate numerator and denominator

First, calculate $\log_{10}\frac{1}{8}$. Since $\frac{1}{8}=8^{-1}=2^{-3}$, $\log_{10}\frac{1}{8}=\log_{10}8^{-1}=-\log_{10}8\approx - 0.90309$.
Then, calculate $\log_{10}6\approx0.77815$.

Step3: Divide numerator by denominator

Now, divide the two results: $\frac{- 0.90309}{0.77815}\approx - 1.1605$. Rounding to the nearest thousandth gives - 1.161.

Answer:

-1.161