Find the Exact Value log base 4 of 20- log base 4 of 45+ log base 4 of 144

$\log_{4} (20) - \log_{4} (45) + \log_{4} (144)$

Step 1

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

$\log_{4} (\frac{20}{45}) + \log_{4} (144)$

Step 2

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

$\log_{4} (\frac{20}{45} \cdot 144)$

Step 3

Cancel the common factor of $9$ .

Factor $9$ out of $45$ .

$\log_{4} (\frac{20}{9 (5)} \cdot 144)$

Factor $9$ out of $144$ .

$\log_{4} (\frac{20}{9 \cdot 5} \cdot (9 \cdot 16))$

Cancel the common factor.

$\log_{4} (\frac{20}{9 \cdot 5} \cdot (9 \cdot 16))$

Rewrite the expression.

$\log_{4} (\frac{20}{5} \cdot 16)$

Step 4

Combine $\frac{20}{5}$ and $16$ .

$\log_{4} (\frac{20 \cdot 16}{5})$

Step 5

Simplify the expression.

Multiply $20$ by $16$ .

$\log_{4} (\frac{320}{5})$

Logarithm base $4$ of $\frac{320}{5}$ is $3$ .

$3$