Simplify/Condense 5 log base 4 of 2+7 log base 4 of x+4 log base 4 of y

$5 \log_{4} (2) + 7 \log_{4} (x) + 4 \log_{4} (y)$

Step 1

Simplify each term.

Logarithm base $4$ of $2$ is $\frac{1}{2}$ .

$5 (\frac{1}{2}) + 7 \log_{4} (x) + 4 \log_{4} (y)$

Combine $5$ and $\frac{1}{2}$ .

$\frac{5}{2} + 7 \log_{4} (x) + 4 \log_{4} (y)$

Simplify $7 \log_{4} (x)$ by moving $7$ inside the logarithm.

$\frac{5}{2} + \log_{4} (x^{7}) + 4 \log_{4} (y)$

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

$\frac{5}{2} + \log_{4} (x^{7}) + \log_{4} (y^{4})$

Step 2

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

$\frac{5}{2} + \log_{4} (x^{7} y^{4})$