Precalculus Examples

$\log_{5} (\sqrt{50})$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{50}$ as $50^{\frac{1}{2}}$ .

$\log_{5} (50^{\frac{1}{2}})$

Step 2

Expand $\log_{5} (50^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\frac{1}{2} \log_{5} (50)$

Step 3

Rewrite $\log_{5} (50)$ as $\log_{5} (2 \cdot 5^{2})$ .

$\frac{1}{2} \log_{5} (2 \cdot 5^{2})$

Step 4

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

$\frac{1}{2} (\log_{5} (2) + \log_{5} (5^{2}))$

Step 5

Use logarithm rules to move $2$ out of the exponent.

$\frac{1}{2} (\log_{5} (2) + 2 \log_{5} (5))$

Step 6

Logarithm base $5$ of $5$ is $1$ .

$\frac{1}{2} (\log_{5} (2) + 2 \cdot 1)$

Step 7

Multiply $2$ by $1$ .

$\frac{1}{2} (\log_{5} (2) + 2)$

Step 8

Apply the distributive property.

$\frac{1}{2} \log_{5} (2) + \frac{1}{2} \cdot 2$

Step 9

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

$\frac{\log_{5} (2)}{2} + \frac{1}{2} \cdot 2$

Step 10

Cancel the common factor of $2$ .

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Step 10.1

Cancel the common factor.

$\frac{\log_{5} (2)}{2} + \frac{1}{2} \cdot 2$

Step 10.2

Rewrite the expression.

$\frac{\log_{5} (2)}{2} + 1$