Algebra Examples

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

Step 1

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

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

Step 2

Multiply $32$ by $162$ .

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

Step 3

Simplify each term.

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Step 3.1

Rewrite $\log_{12} (3)$ using the change of base formula.

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Step 3.1.1

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 $0$ .

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

Step 3.1.2

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

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

Step 3.2

Rewrite $\log_{12} (5184)$ using the change of base formula.

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Step 3.2.1

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 $0$ .

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

Step 3.2.2

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

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

Step 4

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$3$