Precalculus Examples

$\log_{15} (\frac{4 \sqrt{x}}{y})$

Step 1

Rewrite $\log_{15} (\frac{4 \sqrt{x}}{y})$ as $\log_{15} (4 \sqrt{x}) - \log_{15} (y)$ .

$\log_{15} (4 \sqrt{x}) - \log_{15} (y)$

Step 2

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

$\log_{15} (4 x^{\frac{1}{2}}) - \log_{15} (y)$

Step 3

Rewrite $\log_{15} (4 x^{\frac{1}{2}})$ as $\log_{15} (4) + \log_{15} (x^{\frac{1}{2}})$ .

$\log_{15} (4) + \log_{15} (x^{\frac{1}{2}}) - \log_{15} (y)$

Step 4

Expand $\log_{15} (x^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\log_{15} (4) + \frac{1}{2} \log_{15} (x) - \log_{15} (y)$

Step 5

Simplify each term.

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Step 5.1

Rewrite $\log_{15} (4)$ as $\log_{15} (2^{2})$ .

$\log_{15} (2^{2}) + \frac{1}{2} \log_{15} (x) - \log_{15} (y)$

Step 5.2

Expand $\log_{15} (2^{2})$ by moving $2$ outside the logarithm.

$2 \log_{15} (2) + \frac{1}{2} \log_{15} (x) - \log_{15} (y)$

Step 5.3

Combine $\frac{1}{2}$ and $\log_{15} (x)$ .

$2 \log_{15} (2) + \frac{\log_{15} (x)}{2} - \log_{15} (y)$