Write as a Single Logarithm 4 log base 4 of x+5 log base 4 of y-2 log base 4 of z

$4 \log_{4} (x) + 5 \log_{4} (y) - 2 \log_{4} (z)$

Step 1

Simplify each term.

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

$\log_{4} (x^{4}) + 5 \log_{4} (y) - 2 \log_{4} (z)$

Simplify $5 \log_{4} (y)$ by moving $5$ inside the logarithm.

$\log_{4} (x^{4}) + \log_{4} (y^{5}) - 2 \log_{4} (z)$

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

$\log_{4} (x^{4}) + \log_{4} (y^{5}) - \log_{4} (z^{2})$

Step 2

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

$\log_{4} (x^{4} y^{5}) - \log_{4} (z^{2})$

Step 3

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

$\log_{4} (\frac{x^{4} y^{5}}{z^{2}})$