Precalculus Examples

$\log_{7} (\frac{\sqrt{7 x^{5}}}{y})$

Step 1

Rewrite $\log_{7} (\frac{\sqrt{7 x^{5}}}{y})$ as $\log_{7} (\sqrt{7 x^{5}}) - \log_{7} (y)$ .

$\log_{7} (\sqrt{7 x^{5}}) - \log_{7} (y)$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7 x^{5}}$ as ${(7 x^{5})}^{\frac{1}{2}}$ .

$\log_{7} ({(7 x^{5})}^{\frac{1}{2}}) - \log_{7} (y)$

Step 3

Expand $\log_{7} ({(7 x^{5})}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\frac{1}{2} \log_{7} (7 x^{5}) - \log_{7} (y)$

Step 4

Rewrite $\log_{7} (7 x^{5})$ as $\log_{7} (7) + \log_{7} (x^{5})$ .

$\frac{1}{2} (\log_{7} (7) + \log_{7} (x^{5})) - \log_{7} (y)$

Step 5

Expand $\log_{7} (x^{5})$ by moving $5$ outside the logarithm.

$\frac{1}{2} (\log_{7} (7) + 5 \log_{7} (x)) - \log_{7} (y)$

Step 6

Logarithm base $7$ of $7$ is $1$ .

$\frac{1}{2} (1 + 5 \log_{7} (x)) - \log_{7} (y)$

Step 7

Simplify each term.

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Step 7.1

Apply the distributive property.

$\frac{1}{2} \cdot 1 + \frac{1}{2} (5 \log_{7} (x)) - \log_{7} (y)$

Step 7.2

Multiply $\frac{1}{2}$ by $1$ .

$\frac{1}{2} + \frac{1}{2} (5 \log_{7} (x)) - \log_{7} (y)$

Step 7.3

Multiply $\frac{1}{2} (5 \log_{7} (x))$ .

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Step 7.3.1

Combine $5$ and $\frac{1}{2}$ .

$\frac{1}{2} + \frac{5}{2} \log_{7} (x) - \log_{7} (y)$

Step 7.3.2

Combine $\frac{5}{2}$ and $\log_{7} (x)$ .

$\frac{1}{2} + \frac{5 \log_{7} (x)}{2} - \log_{7} (y)$