Simplify/Condense log base 4 of W-1/2( log base 4 of x+5 log base 4 of y)+2 log base 4 of z

$\log_{4} (W) - \frac{1}{2} (\log_{4} (x) + 5 \log_{4} (y)) + 2 \log_{4} (z)$

Step 1

Simplify each term.

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

$\log_{4} (W) - \frac{1}{2} \cdot (\log_{4} (x) + \log_{4} (y^{5})) + 2 \log_{4} (z)$

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

$\log_{4} (W) - \frac{1}{2} \cdot \log_{4} (x y^{5}) + 2 \log_{4} (z)$

Multiply $- \frac{1}{2} \log_{4} (x y^{5})$ .

Reorder $\log_{4} (x y^{5})$ and $\frac{1}{2}$ .

$\log_{4} (W) - (\frac{1}{2} \log_{4} (x y^{5})) + 2 \log_{4} (z)$

Simplify $\frac{1}{2} \log_{4} (x y^{5})$ by moving $\frac{1}{2}$ inside the logarithm.

$\log_{4} (W) - \log_{4} ({(x y^{5})}^{\frac{1}{2}}) + 2 \log_{4} (z)$

Apply the product rule to $x y^{5}$ .

$\log_{4} (W) - \log_{4} (x^{\frac{1}{2}} {(y^{5})}^{\frac{1}{2}}) + 2 \log_{4} (z)$

Multiply the exponents in ${(y^{5})}^{\frac{1}{2}}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\log_{4} (W) - \log_{4} (x^{\frac{1}{2}} y^{5 (\frac{1}{2})}) + 2 \log_{4} (z)$

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

$\log_{4} (W) - \log_{4} (x^{\frac{1}{2}} y^{\frac{5}{2}}) + 2 \log_{4} (z)$

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

$\log_{4} (W) - \log_{4} (x^{\frac{1}{2}} y^{\frac{5}{2}}) + \log_{4} (z^{2})$

Step 2

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

$\log_{4} (\frac{W}{x^{\frac{1}{2}} y^{\frac{5}{2}}}) + \log_{4} (z^{2})$

Step 3

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

$\log_{4} (\frac{W}{x^{\frac{1}{2}} y^{\frac{5}{2}}} z^{2})$

Step 4

Combine $\frac{W}{x^{\frac{1}{2}} y^{\frac{5}{2}}}$ and $z^{2}$ .

$\log_{4} (\frac{W z^{2}}{x^{\frac{1}{2}} y^{\frac{5}{2}}})$