Precalculus Examples

$\log (z + 3) - 2 \log (z + 5) + 5 (\log (z) - \log (x + 3))$

Step 1

Simplify each term.

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Step 1.1

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

$\log (z + 3) - \log ({(z + 5)}^{2}) + 5 (\log (z) - \log (x + 3))$

Step 1.2

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

$\log (z + 3) - \log ({(z + 5)}^{2}) + 5 \log (\frac{z}{x + 3})$

Step 1.3

Simplify $5 \log (\frac{z}{x + 3})$ by moving $5$ inside the logarithm.

$\log (z + 3) - \log ({(z + 5)}^{2}) + \log ({(\frac{z}{x + 3})}^{5})$

Step 1.4

Apply the product rule to $\frac{z}{x + 3}$ .

$\log (z + 3) - \log ({(z + 5)}^{2}) + \log (\frac{z^{5}}{{(x + 3)}^{5}})$

Step 2

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

$\log (\frac{z + 3}{{(z + 5)}^{2}}) + \log (\frac{z^{5}}{{(x + 3)}^{5}})$

Step 3

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

$\log (\frac{z + 3}{{(z + 5)}^{2}} \cdot \frac{z^{5}}{{(x + 3)}^{5}})$

Step 4

Multiply $\frac{z + 3}{{(z + 5)}^{2}}$ by $\frac{z^{5}}{{(x + 3)}^{5}}$ .

$\log (\frac{(z + 3) z^{5}}{{(z + 5)}^{2} {(x + 3)}^{5}})$

Step 5

Reorder factors in $\log (\frac{(z + 3) z^{5}}{{(z + 5)}^{2} {(x + 3)}^{5}})$ .

$\log (\frac{z^{5} (z + 3)}{{(z + 5)}^{2} {(x + 3)}^{5}})$