Precalculus Examples

$\log (\sqrt{10000 y})$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{10000 y}$ as ${(10000 y)}^{\frac{1}{2}}$ .

$\log ({(10000 y)}^{\frac{1}{2}})$

Step 2

Expand $\log ({(10000 y)}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\frac{1}{2} \log (10000 y)$

Step 3

Rewrite $\log (10000 y)$ as $\log (10000) + \log (y)$ .

$\frac{1}{2} (\log (10000) + \log (y))$

Step 4

Logarithm base $10$ of $10000$ is $4$ .

$\frac{1}{2} (4 + \log (y))$

Step 5

Apply the distributive property.

$\frac{1}{2} \cdot 4 + \frac{1}{2} \log (y)$

Step 6

Cancel the common factor of $2$ .

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Step 6.1

Factor $2$ out of $4$ .

$\frac{1}{2} \cdot (2 (2)) + \frac{1}{2} \log (y)$

Step 6.2

Cancel the common factor.

$\frac{1}{2} \cdot (2 \cdot 2) + \frac{1}{2} \log (y)$

Step 6.3

Rewrite the expression.

$2 + \frac{1}{2} \log (y)$

Step 7

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

$2 + \frac{\log (y)}{2}$