Simplify/Condense 2 log base 4 of 20-2 log base 4 of 12

$2 \log_{4} (20) - 2 \log_{4} (12)$

Simplify each term.

Simplify $2 \log_{4} (20)$ by moving $2$ inside the logarithm.

$\log_{4} (20^{2}) - 2 \log_{4} (12)$

Raise $20$ to the power of $2$ .

$\log_{4} (400) - 2 \log_{4} (12)$

Simplify $- 2 \log_{4} (12)$ by moving $2$ inside the logarithm.

$\log_{4} (400) - \log_{4} (12^{2})$

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

$\log_{4} (400) - \log_{4} (144)$

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

$\log_{4} (\frac{400}{144})$

Cancel the common factor of $400$ and $144$ .

Factor $16$ out of $400$ .

$\log_{4} (\frac{16 (25)}{144})$

Cancel the common factors.

Factor $16$ out of $144$ .

$\log_{4} (\frac{16 \cdot 25}{16 \cdot 9})$

Cancel the common factor.

$\log_{4} (\frac{16 \cdot 25}{16 \cdot 9})$

Rewrite the expression.

$\log_{4} (\frac{25}{9})$

The result can be shown in multiple forms.

Exact Form:

$\log_{4} (\frac{25}{9})$

Decimal Form:

$0.73696559 \dots$