Evaluate ( log base 5 of 12- log base 5 of 4)* log base 1/9 of cube root of 5

$(\log_{5} (12) - \log_{5} (4)) \cdot \log_{\frac{1}{9}} (\sqrt[3]{5})$

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\log_{5} (\frac{12}{4}) \cdot \log_{\frac{1}{9}} (\sqrt[3]{5})$

Divide $12$ by $4$ .

$\log_{5} (3) \log_{\frac{1}{9}} (\sqrt[3]{5})$

Rewrite $\log_{5} (3)$ using the change of base formula.

The change of base rule can be used if $a$ and $b$ are greater than $0$ and not equal to $1$ , and $x$ is greater than $0$ .

$\log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)} \log_{\frac{1}{9}} (\sqrt[3]{5})$

Substitute in values for the variables in the change of base formula, using $b = 10$ .

$\frac{\log (3)}{\log (5)} \log_{\frac{1}{9}} (\sqrt[3]{5})$

Combine $\frac{\log (3)}{\log (5)}$ and $\log_{\frac{1}{9}} (\sqrt[3]{5})$ .

$\frac{\log (3) \log_{\frac{1}{9}} (\sqrt[3]{5})}{\log (5)}$

The result can be shown in multiple forms.

Exact Form:

$\frac{\log (3) \log_{\frac{1}{9}} (\sqrt[3]{5})}{\log (5)}$

Decimal Form:

$- 0.1\overline{6}$