Precalculus Examples

$\log_{7} (\frac{\sqrt{x^{2} y^{3}}}{343})$

Step 1

Rewrite $\log_{7} (\frac{\sqrt{x^{2} y^{3}}}{343})$ as $\log_{7} (\sqrt{x^{2} y^{3}}) - \log_{7} (343)$ .

$\log_{7} (\sqrt{x^{2} y^{3}}) - \log_{7} (343)$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{2} y^{3}}$ as ${(x^{2} y^{3})}^{\frac{1}{2}}$ .

$\log_{7} ({(x^{2} y^{3})}^{\frac{1}{2}}) - \log_{7} (343)$

Step 3

Expand $\log_{7} ({(x^{2} y^{3})}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\frac{1}{2} \log_{7} (x^{2} y^{3}) - \log_{7} (343)$

Step 4

Logarithm base $7$ of $343$ is $3$ .

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

Step 5

Multiply $- 1$ by $3$ .

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

Step 6

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

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

Step 7

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

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

Step 8

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

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

Step 9

Simplify each term.

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Step 9.1

Apply the distributive property.

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

Step 9.2

Cancel the common factor of $2$ .

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Step 9.2.1

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

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

Step 9.2.2

Cancel the common factor.

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

Step 9.2.3

Rewrite the expression.

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

Step 9.3

Multiply $\frac{1}{2} (3 \log_{7} (y))$ .

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Step 9.3.1

Combine $3$ and $\frac{1}{2}$ .

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

Step 9.3.2

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

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