Evaluate - log base 12 of 3+ log base 12 of 32+ log base 12 of 162

$- \log_{12} (3) + \log_{12} (32) + \log_{12} (162)$

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

$- \log_{12} (3) + \log_{12} (32 \cdot 162)$

Multiply $32$ by $162$ .

$- \log_{12} (3) + \log_{12} (5184)$

Simplify each term.

Rewrite $\log_{12} (3)$ 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 $0$ .

$- \log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)} + \log_{12} (5184)$

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

$- \frac{\log (3)}{\log (12)} + \log_{12} (5184)$

Rewrite $\log_{12} (5184)$ 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 $0$ .

$- \frac{\log (3)}{\log (12)} + \log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)}$

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

$- \frac{\log (3)}{\log (12)} + \frac{\log (5184)}{\log (12)}$

The result can be shown in multiple forms.

Exact Form:

$- \frac{\log (3)}{\log (12)} + \frac{\log (5184)}{\log (12)}$

Decimal Form:

$3$