Precalculus Examples

$\log (\frac{\sqrt{x}}{100000 y^{6}})$

Step 1

Rewrite $\log (\frac{\sqrt{x}}{100000 y^{6}})$ as $\log (\sqrt{x}) - \log (100000 y^{6})$ .

$\log (\sqrt{x}) - \log (100000 y^{6})$

Step 2

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

$\log (x^{\frac{1}{2}}) - \log (100000 y^{6})$

Step 3

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

$\frac{1}{2} \log (x) - \log (100000 y^{6})$

Step 4

Rewrite $\log (100000 y^{6})$ as $\log (100000) + \log (y^{6})$ .

$\frac{1}{2} \log (x) - (\log (100000) + \log (y^{6}))$

Step 5

Expand $\log (y^{6})$ by moving $6$ outside the logarithm.

$\frac{1}{2} \log (x) - (\log (100000) + 6 \log (y))$

Step 6

Logarithm base $10$ of $100000$ is $5$ .

$\frac{1}{2} \log (x) - (5 + 6 \log (y))$

Step 7

Simplify each term.

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Step 7.1

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

$\frac{\log (x)}{2} - (5 + 6 \log (y))$

Step 7.2

Apply the distributive property.

$\frac{\log (x)}{2} - 1 \cdot 5 - (6 \log (y))$

Step 7.3

Multiply $- 1$ by $5$ .

$\frac{\log (x)}{2} - 5 - (6 \log (y))$

Step 7.4

Multiply $6$ by $- 1$ .

$\frac{\log (x)}{2} - 5 - 6 \log (y)$