Precalculus Examples

$\log_{7} (\frac{\sqrt[7]{x^{4} y}}{49})$

Step 1

Rewrite $\log_{7} (\frac{\sqrt[7]{x^{4} y}}{49})$ as $\log_{7} (\sqrt[7]{x^{4} y}) - \log_{7} (49)$ .

$\log_{7} (\sqrt[7]{x^{4} y}) - \log_{7} (49)$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[7]{x^{4} y}$ as ${(x^{4} y)}^{\frac{1}{7}}$ .

$\log_{7} ({(x^{4} y)}^{\frac{1}{7}}) - \log_{7} (49)$

Step 3

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

$\frac{1}{7} \log_{7} (x^{4} y) - \log_{7} (49)$

Step 4

Logarithm base $7$ of $49$ is $2$ .

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

Step 5

Multiply $- 1$ by $2$ .

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

Step 6

Rewrite $\log_{7} (x^{4} y)$ as $\log_{7} (x^{4}) + \log_{7} (y)$ .

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

Step 7

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

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

Step 8

Simplify each term.

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Step 8.1

Apply the distributive property.

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

Step 8.2

Multiply $\frac{1}{7} (4 \log_{7} (x))$ .

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Step 8.2.1

Combine $4$ and $\frac{1}{7}$ .

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

Step 8.2.2

Combine $\frac{4}{7}$ and $\log_{7} (x)$ .

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

Step 8.3

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

$\frac{4 \log_{7} (x)}{7} + \frac{\log_{7} (y)}{7} - 2$