Evaluate log base 3 of b* log base b of 27

$\log_{3} (b) \cdot \log_{b} (27)$

Rewrite $\log_{3} (b)$ 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 $0$ .

$\log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)} \log_{b} (27)$

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

$\frac{\log (b)}{\log (3)} \log_{b} (27)$

Rewrite $\log_{b} (27)$ 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 $0$ .

$\frac{\log (b)}{\log (3)} \log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)}$

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

$\frac{\log (b)}{\log (3)} \cdot \frac{\log (27)}{\log (b)}$

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

Cancel the common factor.

$\frac{\log (b)}{\log (3)} \cdot \frac{\log (27)}{\log (b)}$

Rewrite the expression.

$\frac{1}{\log (3)} \log (27)$

Combine $\frac{1}{\log (3)}$ and $\log (27)$ .

$\frac{\log (27)}{\log (3)}$

Rewrite $\log (27)$ as $\log (3^{3})$ .

$\frac{\log (3^{3})}{\log (3)}$

Expand $\log (3^{3})$ by moving $3$ outside the logarithm.

$\frac{3 \log (3)}{\log (3)}$

Cancel the common factor of $\log (3)$ .

Cancel the common factor.

$\frac{3 \log (3)}{\log (3)}$

Divide $3$ by $1$ .

$3$