Precalculus Examples

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

Step 1

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

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

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

$\log_{a} (y^{4} x^{\frac{1}{2}}) - \log_{a} (w z^{4})$

Step 3

Rewrite $\log_{a} (y^{4} x^{\frac{1}{2}})$ as $\log_{a} (y^{4}) + \log_{a} (x^{\frac{1}{2}})$ .

$\log_{a} (y^{4}) + \log_{a} (x^{\frac{1}{2}}) - \log_{a} (w z^{4})$

Step 4

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

$4 \log_{a} (y) + \log_{a} (x^{\frac{1}{2}}) - \log_{a} (w z^{4})$

Step 5

Expand $\log_{a} (x^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$4 \log_{a} (y) + \frac{1}{2} \log_{a} (x) - \log_{a} (w z^{4})$

Step 6

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

$4 \log_{a} (y) + \frac{1}{2} \log_{a} (x) - (\log_{a} (w) + \log_{a} (z^{4}))$

Step 7

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

$4 \log_{a} (y) + \frac{1}{2} \log_{a} (x) - (\log_{a} (w) + 4 \log_{a} (z))$

Step 8

Simplify each term.

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Step 8.1

Combine $\frac{1}{2}$ and $\log_{a} (x)$ .

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

Step 8.2

Apply the distributive property.

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

Step 8.3

Multiply $4$ by $- 1$ .

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