Precalculus Examples

$\log_{7} (\frac{\sqrt[4]{9}}{y^{2} x})$

Step 1

Rewrite $\log_{7} (\frac{\sqrt[4]{9}}{y^{2} x})$ as $\log_{7} (\sqrt[4]{9}) - \log_{7} (y^{2} x)$ .

$\log_{7} (\sqrt[4]{9}) - \log_{7} (y^{2} x)$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{9}$ as $9^{\frac{1}{4}}$ .

$\log_{7} (9^{\frac{1}{4}}) - \log_{7} (y^{2} x)$

Step 3

Expand $\log_{7} (9^{\frac{1}{4}})$ by moving $\frac{1}{4}$ outside the logarithm.

$\frac{1}{4} \log_{7} (9) - \log_{7} (y^{2} x)$

Step 4

Rewrite $\log_{7} (9)$ as $\log_{7} (3^{2})$ .

$\frac{1}{4} \log_{7} (3^{2}) - \log_{7} (y^{2} x)$

Step 5

Expand $\log_{7} (3^{2})$ by moving $2$ outside the logarithm.

$\frac{1}{4} (2 \log_{7} (3)) - \log_{7} (y^{2} x)$

Step 6

Cancel the common factor of $2$ .

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Step 6.1

Factor $2$ out of $4$ .

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

Step 6.2

Factor $2$ out of $2 \log_{7} (3)$ .

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

Step 6.3

Cancel the common factor.

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

Step 6.4

Rewrite the expression.

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

Step 7

Combine $\frac{1}{2}$ and $\log_{7} (3)$ .

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

Step 8

Rewrite $\log_{7} (y^{2} x)$ as $\log_{7} (y^{2}) + \log_{7} (x)$ .

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

Step 9

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

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

Step 10

Simplify each term.

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Step 10.1

Apply the distributive property.

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

Step 10.2

Multiply $2$ by $- 1$ .

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