Algebra Examples

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

Step 1

Simplify each term.

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Step 1.1

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

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

Step 1.2

Apply the distributive property.

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

Step 1.3

Multiply $z$ by $z$ .

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

Step 1.4

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

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

Step 1.5

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

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

Step 1.6

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

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

Step 2

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

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

Step 3

Simplify the numerator.

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Step 3.1

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

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Step 3.1.1

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

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

Step 3.1.2

Factor $z$ out of $5 z$ .

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

Step 3.1.3

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

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

Step 3.2

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

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