Evaluate log base 4 of m* log base m of 20

$\log_{4} (m) \cdot \log_{m} (20)$

Rewrite $\log_{4} (m)$ 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 $0$ .

$\log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)} \log_{m} (20)$

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

$\frac{\log (m)}{\log (4)} \log_{m} (20)$

Rewrite $\log_{m} (20)$ 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 $0$ .

$\frac{\log (m)}{\log (4)} \log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)}$

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

$\frac{\log (m)}{\log (4)} \cdot \frac{\log (20)}{\log (m)}$

Cancel the common factor of $\log (m)$ .

Cancel the common factor.

$\frac{\log (m)}{\log (4)} \cdot \frac{\log (20)}{\log (m)}$

Rewrite the expression.

$\frac{1}{\log (4)} \log (20)$

Combine $\frac{1}{\log (4)}$ and $\log (20)$ .

$\frac{\log (20)}{\log (4)}$

The result can be shown in multiple forms.

Exact Form:

$\frac{\log (20)}{\log (4)}$

Decimal Form:

$2.16096404 \dots$