Question

use the change of base formula to compute \\(\log_{1/5} \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 $c = 10$ (common logarithm) or $c = e$ (natural logarithm). Let's use common logarithms. So, for $\log_{\frac{1}{5}}\frac{1}{8}$, we have $a=\frac{1}{8}$ and $b = \frac{1}{5}$. So, $\log_{\frac{1}{5}}\frac{1}{8}=\frac{\log_{10}\frac{1}{8}}{\log_{10}\frac{1}{5}}$.

Step2: Calculate Numerator and Denominator

First, calculate $\log_{10}\frac{1}{8}$. We know that $\frac{1}{8}=8^{-1}=2^{-3}$, so $\log_{10}\frac{1}{8}=\log_{10}2^{-3}=- 3\log_{10}2\approx - 3\times0.3010=-0.9030$.

Next, calculate $\log_{10}\frac{1}{5}$. We know that $\frac{1}{5}=5^{-1}$, so $\log_{10}\frac{1}{5}=\log_{10}5^{-1}=-\log_{10}5\approx - 0.6990$.

Step3: Divide the Two Values

Now, divide the numerator by the denominator: $\frac{-0.9030}{-0.6990}\approx1.292$. (We can also use a calculator directly to compute $\log_{10}(\frac{1}{8})$ and $\log_{10}(\frac{1}{5})$ and then divide. Alternatively, using natural logarithm: $\frac{\ln(\frac{1}{8})}{\ln(\frac{1}{5})}=\frac{-\ln(8)}{-\ln(5)}=\frac{\ln(8)}{\ln(5)}$. $\ln(8)\approx2.0794$, $\ln(5)\approx1.6094$, so $\frac{2.0794}{1.6094}\approx1.292$.)

Answer:

1.292