Precalculus Examples

$\log_{5} (3 \sqrt{\frac{x^{2} y}{25}})$

Step 1

Rewrite $\log_{5} (3 \sqrt{\frac{x^{2} y}{25}})$ as $\log_{5} (3) + \log_{5} (\sqrt{\frac{x^{2} y}{25}})$ .

$\log_{5} (3) + \log_{5} (\sqrt{\frac{x^{2} y}{25}})$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{\frac{x^{2} y}{25}}$ as ${(\frac{x^{2} y}{25})}^{\frac{1}{2}}$ .

$\log_{5} (3) + \log_{5} ({(\frac{x^{2} y}{25})}^{\frac{1}{2}})$

Step 3

Expand $\log_{5} ({(\frac{x^{2} y}{25})}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\log_{5} (3) + \frac{1}{2} \log_{5} (\frac{x^{2} y}{25})$

Step 4

Rewrite $\log_{5} (\frac{x^{2} y}{25})$ as $\log_{5} (x^{2} y) - \log_{5} (25)$ .

$\log_{5} (3) + \frac{1}{2} (\log_{5} (x^{2} y) - \log_{5} (25))$

Step 5

Logarithm base $5$ of $25$ is $2$ .

$\log_{5} (3) + \frac{1}{2} (\log_{5} (x^{2} y) - 1 \cdot 2)$

Step 6

Multiply $- 1$ by $2$ .

$\log_{5} (3) + \frac{1}{2} (\log_{5} (x^{2} y) - 2)$

Step 7

Rewrite $\log_{5} (x^{2} y)$ as $\log_{5} (x^{2}) + \log_{5} (y)$ .

$\log_{5} (3) + \frac{1}{2} (\log_{5} (x^{2}) + \log_{5} (y) - 2)$

Step 8

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

$\log_{5} (3) + \frac{1}{2} (2 \log_{5} (x) + \log_{5} (y) - 2)$

Step 9

Simplify each term.

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Step 9.1

Apply the distributive property.

$\log_{5} (3) + \frac{1}{2} (2 \log_{5} (x)) + \frac{1}{2} \log_{5} (y) + \frac{1}{2} \cdot - 2$

Step 9.2

Simplify.

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Step 9.2.1

Cancel the common factor of $2$ .

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Step 9.2.1.1

Factor $2$ out of $2 \log_{5} (x)$ .

$\log_{5} (3) + \frac{1}{2} (2 (\log_{5} (x))) + \frac{1}{2} \log_{5} (y) + \frac{1}{2} \cdot - 2$

Step 9.2.1.2

Cancel the common factor.

$\log_{5} (3) + \frac{1}{2} (2 \log_{5} (x)) + \frac{1}{2} \log_{5} (y) + \frac{1}{2} \cdot - 2$

Step 9.2.1.3

Rewrite the expression.

$\log_{5} (3) + \log_{5} (x) + \frac{1}{2} \log_{5} (y) + \frac{1}{2} \cdot - 2$

Step 9.2.2

Combine $\frac{1}{2}$ and $\log_{5} (y)$ .

$\log_{5} (3) + \log_{5} (x) + \frac{\log_{5} (y)}{2} + \frac{1}{2} \cdot - 2$

Step 9.2.3

Cancel the common factor of $2$ .

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Step 9.2.3.1

Factor $2$ out of $- 2$ .

$\log_{5} (3) + \log_{5} (x) + \frac{\log_{5} (y)}{2} + \frac{1}{2} \cdot (2 (- 1))$

Step 9.2.3.2

Cancel the common factor.

$\log_{5} (3) + \log_{5} (x) + \frac{\log_{5} (y)}{2} + \frac{1}{2} \cdot (2 \cdot - 1)$

Step 9.2.3.3

Rewrite the expression.

$\log_{5} (3) + \log_{5} (x) + \frac{\log_{5} (y)}{2} - 1$