Precalculus Examples

$\log_{7} (\frac{6 \sqrt{x}}{y})$

Step 1

Rewrite $\log_{7} (\frac{6 \sqrt{x}}{y})$ as $\log_{7} (6 \sqrt{x}) - \log_{7} (y)$ .

$\log_{7} (6 \sqrt{x}) - \log_{7} (y)$

Step 2

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

$\log_{7} (6 x^{\frac{1}{2}}) - \log_{7} (y)$

Step 3

Rewrite $\log_{7} (6 x^{\frac{1}{2}})$ as $\log_{7} (6) + \log_{7} (x^{\frac{1}{2}})$ .

$\log_{7} (6) + \log_{7} (x^{\frac{1}{2}}) - \log_{7} (y)$

Step 4

Expand $\log_{7} (x^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\log_{7} (6) + \frac{1}{2} \log_{7} (x) - \log_{7} (y)$

Step 5

Simplify each term.

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Step 5.1

Rewrite $\log_{7} (6)$ as $\log_{7} (2 (3))$ .

$\log_{7} (2 (3)) + \frac{1}{2} \log_{7} (x) - \log_{7} (y)$

Step 5.2

Rewrite $\log_{7} (2 (3))$ as $\log_{7} (2) + \log_{7} (3)$ .

$\log_{7} (2) + \log_{7} (3) + \frac{1}{2} \log_{7} (x) - \log_{7} (y)$

Step 5.3

Combine $\frac{1}{2}$ and $\log_{7} (x)$ .

$\log_{7} (2) + \log_{7} (3) + \frac{\log_{7} (x)}{2} - \log_{7} (y)$