Evaluate ( log base 5 of m)/( log base 15 of m)

$\frac{\log_{5} (m)}{\log_{15} (m)}$

Rewrite $\log_{5} (m)$ 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 $0$ .

$\frac{\log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)}}{\log_{15} (m)}$

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

$\frac{\frac{\log (m)}{\log (5)}}{\log_{15} (m)}$

Rewrite $\log_{15} (m)$ 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 $0$ .

$\frac{\frac{\log (m)}{\log (5)}}{\log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)}}$

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

$\frac{\frac{\log (m)}{\log (5)}}{\frac{\log (m)}{\log (15)}}$

Multiply the numerator by the reciprocal of the denominator.

$\frac{\log (m)}{\log (5)} \cdot \frac{\log (15)}{\log (m)}$

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

Cancel the common factor.

$\frac{\log (m)}{\log (5)} \cdot \frac{\log (15)}{\log (m)}$

Rewrite the expression.

$\frac{1}{\log (5)} \log (15)$

Combine $\frac{1}{\log (5)}$ and $\log (15)$ .

$\frac{\log (15)}{\log (5)}$

The result can be shown in multiple forms.

Exact Form:

$\frac{\log (15)}{\log (5)}$

Decimal Form:

$1.68260619 \dots$