Precalculus Examples

$\log_{5} (\frac{x^{4}}{y^{4} z^{9}})$

Step 1

Rewrite $\log_{5} (\frac{x^{4}}{y^{4} z^{9}})$ as $\log_{5} (x^{4}) - \log_{5} (y^{4} z^{9})$ .

$\log_{5} (x^{4}) - \log_{5} (y^{4} z^{9})$

Step 2

Expand $\log_{5} (x^{4})$ by moving $4$ outside the logarithm.

$4 \log_{5} (x) - \log_{5} (y^{4} z^{9})$

Step 3

Rewrite $\log_{5} (y^{4} z^{9})$ as $\log_{5} (y^{4}) + \log_{5} (z^{9})$ .

$4 \log_{5} (x) - (\log_{5} (y^{4}) + \log_{5} (z^{9}))$

Step 4

Expand $\log_{5} (y^{4})$ by moving $4$ outside the logarithm.

$4 \log_{5} (x) - (4 \log_{5} (y) + \log_{5} (z^{9}))$

Step 5

Expand $\log_{5} (z^{9})$ by moving $9$ outside the logarithm.

$4 \log_{5} (x) - (4 \log_{5} (y) + 9 \log_{5} (z))$

Step 6

Simplify each term.

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Step 6.1

Apply the distributive property.

$4 \log_{5} (x) - (4 \log_{5} (y)) - (9 \log_{5} (z))$

Step 6.2

Multiply $4$ by $- 1$ .

$4 \log_{5} (x) - 4 \log_{5} (y) - (9 \log_{5} (z))$

Step 6.3

Multiply $9$ by $- 1$ .

$4 \log_{5} (x) - 4 \log_{5} (y) - 9 \log_{5} (z)$