Precalculus Examples

$\log (\frac{\sqrt{x}}{1000 y^{2}})$

Step 1

Rewrite $\log (\frac{\sqrt{x}}{1000 y^{2}})$ as $\log (\sqrt{x}) - \log (1000 y^{2})$ .

$\log (\sqrt{x}) - \log (1000 y^{2})$

Step 2

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

$\log (x^{\frac{1}{2}}) - \log (1000 y^{2})$

Step 3

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

$\frac{1}{2} \log (x) - \log (1000 y^{2})$

Step 4

Rewrite $\log (1000 y^{2})$ as $\log (1000) + \log (y^{2})$ .

$\frac{1}{2} \log (x) - (\log (1000) + \log (y^{2}))$

Step 5

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

$\frac{1}{2} \log (x) - (\log (1000) + 2 \log (y))$

Step 6

Logarithm base $10$ of $1000$ is $3$ .

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

Step 7

Simplify each term.

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Step 7.1

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

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

Step 7.2

Apply the distributive property.

$\frac{\log (x)}{2} - 1 \cdot 3 - (2 \log (y))$

Step 7.3

Multiply $- 1$ by $3$ .

$\frac{\log (x)}{2} - 3 - (2 \log (y))$

Step 7.4

Multiply $2$ by $- 1$ .

$\frac{\log (x)}{2} - 3 - 2 \log (y)$