Precalculus Examples

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

Step 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 2

Simplify each term.

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Step 2.1

Apply the distributive property.

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

Step 2.2

Multiply $z$ by $z$ .

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

Step 2.3

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

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

Step 2.4

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

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

Step 3

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

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

Step 4

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

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Step 4.1

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

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

Step 4.2

Factor $z$ out of $5 z$ .

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

Step 4.3

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

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

Step 5

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

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

Step 6

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 6.1

Apply the product rule to $\frac{z (z + 5)}{{(z - 5)}^{2}}$ .

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

Step 6.2

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

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

Step 7

Multiply the exponents in ${({(z - 5)}^{2})}^{4}$ .

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Step 7.1

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

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

Step 7.2

Multiply $2$ by $4$ .

$\ln (\frac{z^{4} {(z + 5)}^{4}}{{(z - 5)}^{8}})$