Precalculus Examples

$\log_{x} (\sqrt{\frac{57}{74}})$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{\frac{57}{74}}$ as ${(\frac{57}{74})}^{\frac{1}{2}}$ .

$\log_{x} ({(\frac{57}{74})}^{\frac{1}{2}})$

Step 2

Expand $\log_{x} ({(\frac{57}{74})}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\frac{1}{2} \log_{x} (\frac{57}{74})$

Step 3

Rewrite $\log_{x} (\frac{57}{74})$ as $\log_{x} (57) - \log_{x} (74)$ .

$\frac{1}{2} (\log_{x} (57) - \log_{x} (74))$

Step 4

Simplify each term.

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Step 4.1

Rewrite $\log_{x} (57)$ as $\log_{x} (3 (19))$ .

$\frac{1}{2} (\log_{x} (3 (19)) - \log_{x} (74))$

Step 4.2

Rewrite $\log_{x} (3 (19))$ as $\log_{x} (3) + \log_{x} (19)$ .

$\frac{1}{2} (\log_{x} (3) + \log_{x} (19) - \log_{x} (74))$

Step 4.3

Rewrite $\log_{x} (74)$ as $\log_{x} (2 (37))$ .

$\frac{1}{2} (\log_{x} (3) + \log_{x} (19) - \log_{x} (2 (37)))$

Step 4.4

Rewrite $\log_{x} (2 (37))$ as $\log_{x} (2) + \log_{x} (37)$ .

$\frac{1}{2} (\log_{x} (3) + \log_{x} (19) - (\log_{x} (2) + \log_{x} (37)))$

Step 4.5

Apply the distributive property.

$\frac{1}{2} (\log_{x} (3) + \log_{x} (19) - \log_{x} (2) - \log_{x} (37))$

Step 5

Apply the distributive property.

$\frac{1}{2} \log_{x} (3) + \frac{1}{2} \log_{x} (19) + \frac{1}{2} (- \log_{x} (2)) + \frac{1}{2} (- \log_{x} (37))$

Step 6

Simplify.

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Step 6.1

Combine $\frac{1}{2}$ and $\log_{x} (3)$ .

$\frac{\log_{x} (3)}{2} + \frac{1}{2} \log_{x} (19) + \frac{1}{2} (- \log_{x} (2)) + \frac{1}{2} (- \log_{x} (37))$

Step 6.2

Combine $\frac{1}{2}$ and $\log_{x} (19)$ .

$\frac{\log_{x} (3)}{2} + \frac{\log_{x} (19)}{2} + \frac{1}{2} (- \log_{x} (2)) + \frac{1}{2} (- \log_{x} (37))$

Step 6.3

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

$\frac{\log_{x} (3)}{2} + \frac{\log_{x} (19)}{2} - \frac{\log_{x} (2)}{2} + \frac{1}{2} (- \log_{x} (37))$

Step 6.4

Combine $\frac{1}{2}$ and $\log_{x} (37)$ .

$\frac{\log_{x} (3)}{2} + \frac{\log_{x} (19)}{2} - \frac{\log_{x} (2)}{2} - \frac{\log_{x} (37)}{2}$