Evaluate 2 log base 5 of 12- log base 5 of 8-2 log base 5 of 3

$2 \log_{5} (12) - \log_{5} (8) - 2 \log_{5} (3)$

Simplify each term.

Simplify $2 \log_{5} (12)$ by moving $2$ inside the logarithm.

$\log_{5} (12^{2}) - \log_{5} (8) - 2 \log_{5} (3)$

Raise $12$ to the power of $2$ .

$\log_{5} (144) - \log_{5} (8) - 2 \log_{5} (3)$

Simplify $- 2 \log_{5} (3)$ by moving $2$ inside the logarithm.

$\log_{5} (144) - \log_{5} (8) - \log_{5} (3^{2})$

Raise $3$ to the power of $2$ .

$\log_{5} (144) - \log_{5} (8) - \log_{5} (9)$

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\log_{5} (\frac{144}{8}) - \log_{5} (9)$

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\log_{5} (\frac{\frac{144}{8}}{9})$

Cancel the common factor of $144$ and $8$ .

Factor $8$ out of $144$ .

$\log_{5} (\frac{\frac{8 \cdot 18}{8}}{9})$

Cancel the common factors.

Factor $8$ out of $8$ .

$\log_{5} (\frac{\frac{8 \cdot 18}{8 (1)}}{9})$

Cancel the common factor.

$\log_{5} (\frac{\frac{8 \cdot 18}{8 \cdot 1}}{9})$

Rewrite the expression.

$\log_{5} (\frac{\frac{18}{1}}{9})$

Divide $18$ by $1$ .

$\log_{5} (\frac{18}{9})$

Cancel the common factor of $18$ and $9$ .

Factor $9$ out of $18$ .

$\log_{5} (\frac{9 \cdot 2}{9})$

Cancel the common factors.

Factor $9$ out of $9$ .

$\log_{5} (\frac{9 \cdot 2}{9 (1)})$

Cancel the common factor.

$\log_{5} (\frac{9 \cdot 2}{9 \cdot 1})$

Rewrite the expression.

$\log_{5} (\frac{2}{1})$

Divide $2$ by $1$ .

$\log_{5} (2)$

Rewrite $\log_{5} (2)$ 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 $0$ .

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

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

$\frac{\log (2)}{\log (5)}$

The result can be shown in multiple forms.

Exact Form:

$\frac{\log (2)}{\log (5)}$

Decimal Form:

$0.43067655 \dots$

Evaluate 2 log base 5 of 12- log base 5 of 8-2 log base 5 of 3

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