Simplify/Condense 8( log base z of p- log base z of q)-(1/3 log base z of u-3-4 log base z of v)

$8 (\log_{z} (p) - \log_{z} (q)) - (\frac{1}{3} \log_{z} (u - 3) - 4 \log_{z} (v))$

Step 1

Simplify each term.

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

$8 \log_{z} (\frac{p}{q}) - (\frac{1}{3} \cdot \log_{z} (u - 3) - 4 \log_{z} (v))$

Simplify $8 \log_{z} (\frac{p}{q})$ by moving $8$ inside the logarithm.

$\log_{z} ({(\frac{p}{q})}^{8}) - (\frac{1}{3} \cdot \log_{z} (u - 3) - 4 \log_{z} (v))$

Apply the product rule to $\frac{p}{q}$ .

$\log_{z} (\frac{p^{8}}{q^{8}}) - (\frac{1}{3} \cdot \log_{z} (u - 3) - 4 \log_{z} (v))$

Simplify each term.

Simplify $\frac{1}{3} \log_{z} (u - 3)$ by moving $\frac{1}{3}$ inside the logarithm.

$\log_{z} (\frac{p^{8}}{q^{8}}) - (\log_{z} ({(u - 3)}^{\frac{1}{3}}) - 4 \log_{z} (v))$

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

$\log_{z} (\frac{p^{8}}{q^{8}}) - (\log_{z} ({(u - 3)}^{\frac{1}{3}}) - \log_{z} (v^{4}))$

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

$\log_{z} (\frac{p^{8}}{q^{8}}) - \log_{z} (\frac{{(u - 3)}^{\frac{1}{3}}}{v^{4}})$

Step 2

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

$\log_{z} (\frac{\frac{p^{8}}{q^{8}}}{\frac{{(u - 3)}^{\frac{1}{3}}}{v^{4}}})$

Step 3

Multiply the numerator by the reciprocal of the denominator.

$\log_{z} (\frac{p^{8}}{q^{8}} \cdot \frac{v^{4}}{{(u - 3)}^{\frac{1}{3}}})$

Step 4

Combine.

$\log_{z} (\frac{p^{8} v^{4}}{q^{8} {(u - 3)}^{\frac{1}{3}}})$