Precalculus Examples

$\log_{a} (\frac{8 x y^{3}}{z^{4}})$

Step 1

Rewrite $\log_{a} (\frac{8 x y^{3}}{z^{4}})$ as $\log_{a} (8 x y^{3}) - \log_{a} (z^{4})$ .

$\log_{a} (8 x y^{3}) - \log_{a} (z^{4})$

Step 2

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

$\log_{a} (8 x y^{3}) - (4 \log_{a} (z))$

Step 3

Multiply $4$ by $- 1$ .

$\log_{a} (8 x y^{3}) - 4 \log_{a} (z)$

Step 4

Rewrite $\log_{a} (8 x y^{3})$ as $\log_{a} (8 x) + \log_{a} (y^{3})$ .

$\log_{a} (8 x) + \log_{a} (y^{3}) - 4 \log_{a} (z)$

Step 5

Rewrite $\log_{a} (8 x)$ as $\log_{a} (8) + \log_{a} (x)$ .

$\log_{a} (8) + \log_{a} (x) + \log_{a} (y^{3}) - 4 \log_{a} (z)$

Step 6

Expand $\log_{a} (y^{3})$ by moving $3$ outside the logarithm.

$\log_{a} (8) + \log_{a} (x) + 3 \log_{a} (y) - 4 \log_{a} (z)$

Step 7

Simplify each term.

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Step 7.1

Rewrite $\log_{a} (8)$ as $\log_{a} (2^{3})$ .

$\log_{a} (2^{3}) + \log_{a} (x) + 3 \log_{a} (y) - 4 \log_{a} (z)$

Step 7.2

Expand $\log_{a} (2^{3})$ by moving $3$ outside the logarithm.

$3 \log_{a} (2) + \log_{a} (x) + 3 \log_{a} (y) - 4 \log_{a} (z)$