Write as a Single Logarithm 4( natural log of z+ natural log of z+5)-2 natural log of z-5

$4 (\ln (z) + \ln (z + 5)) - 2 \ln (z - 5)$

Simplify each term.

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

$4 \ln (z (z + 5)) - 2 \ln (z - 5)$

Apply the distributive property.

$4 \ln (z \cdot z + z \cdot 5) - 2 \ln (z - 5)$

Multiply $z$ by $z$ .

$4 \ln (z^{2} + z \cdot 5) - 2 \ln (z - 5)$

Move $5$ to the left of $z$ .

$4 \ln (z^{2} + 5 \cdot z) - 2 \ln (z - 5)$

Simplify $4 \ln (z^{2} + 5 z)$ by moving $4$ inside the logarithm.

$\ln ({(z^{2} + 5 z)}^{4}) - 2 \ln (z - 5)$

Simplify $- 2 \ln (z - 5)$ by moving $2$ inside the logarithm.

$\ln ({(z^{2} + 5 z)}^{4}) - \ln ({(z - 5)}^{2})$

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

$\ln (\frac{{(z^{2} + 5 z)}^{4}}{{(z - 5)}^{2}})$

Simplify the numerator.

Factor $z$ out of $z^{2} + 5 z$ .

Factor $z$ out of $z^{2}$ .

$\ln (\frac{{(z \cdot z + 5 z)}^{4}}{{(z - 5)}^{2}})$

Factor $z$ out of $5 z$ .

$\ln (\frac{{(z \cdot z + z \cdot 5)}^{4}}{{(z - 5)}^{2}})$

Factor $z$ out of $z \cdot z + z \cdot 5$ .

$\ln (\frac{{(z (z + 5))}^{4}}{{(z - 5)}^{2}})$

Apply the product rule to $z (z + 5)$ .

$\ln (\frac{z^{4} {(z + 5)}^{4}}{{(z - 5)}^{2}})$