Precalculus Examples

$8 \log (\sqrt{x - 1}) - \log_{a} (\frac{4}{x} + \log_{a} (4))$

Step 1

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

$8 \log ({(x - 1)}^{\frac{1}{2}}) - \log_{a} (\frac{4}{x} + \log_{a} (4))$

Step 2

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

$8 (\frac{1}{2} \log (x - 1)) - \log_{a} (\frac{4}{x} + \log_{a} (4))$

Step 3

Simplify each term.

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Step 3.1

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

$8 \frac{\log (x - 1)}{2} - \log_{a} (\frac{4}{x} + \log_{a} (4))$

Step 3.2

Cancel the common factor of $2$ .

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Step 3.2.1

Factor $2$ out of $8$ .

$2 (4) \frac{\log (x - 1)}{2} - \log_{a} (\frac{4}{x} + \log_{a} (4))$

Step 3.2.2

Cancel the common factor.

$2 \cdot 4 \frac{\log (x - 1)}{2} - \log_{a} (\frac{4}{x} + \log_{a} (4))$

Step 3.2.3

Rewrite the expression.

$4 \log (x - 1) - \log_{a} (\frac{4}{x} + \log_{a} (4))$

Step 3.3

Simplify each term.

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Step 3.3.1

Rewrite $\log_{a} (4)$ as $\log_{a} (2^{2})$ .

$4 \log (x - 1) - \log_{a} (\frac{4}{x} + \log_{a} (2^{2}))$

Step 3.3.2

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

$4 \log (x - 1) - \log_{a} (\frac{4}{x} + 2 \log_{a} (2))$