Precalculus Examples

$\log_{a} (\frac{x^{3} w}{y^{2} z^{4}})$

Step 1

Rewrite $\log_{a} (\frac{x^{3} w}{y^{2} z^{4}})$ as $\log_{a} (x^{3} w) - \log_{a} (y^{2} z^{4})$ .

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

Step 2

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

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

Step 3

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

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

Step 4

Rewrite $\log_{a} (y^{2} z^{4})$ as $\log_{a} (y^{2}) + \log_{a} (z^{4})$ .

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

Step 5

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

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

Step 6

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

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

Step 7

Simplify each term.

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Step 7.1

Apply the distributive property.

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

Step 7.2

Multiply $2$ by $- 1$ .

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

Step 7.3

Multiply $4$ by $- 1$ .

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