Precalculus Examples

$\log_{4} (\sqrt{\frac{b^{7}}{b + 3}})$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{\frac{b^{7}}{b + 3}}$ as ${(\frac{b^{7}}{b + 3})}^{\frac{1}{2}}$ .

$\log_{4} ({(\frac{b^{7}}{b + 3})}^{\frac{1}{2}})$

Step 2

Expand $\log_{4} ({(\frac{b^{7}}{b + 3})}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\frac{1}{2} \log_{4} (\frac{b^{7}}{b + 3})$

Step 3

Rewrite $\log_{4} (\frac{b^{7}}{b + 3})$ as $\log_{4} (b^{7}) - \log_{4} (b + 3)$ .

$\frac{1}{2} (\log_{4} (b^{7}) - \log_{4} (b + 3))$

Step 4

Expand $\log_{4} (b^{7})$ by moving $7$ outside the logarithm.

$\frac{1}{2} (7 \log_{4} (b) - \log_{4} (b + 3))$

Step 5

Apply the distributive property.

$\frac{1}{2} (7 \log_{4} (b)) + \frac{1}{2} (- \log_{4} (b + 3))$

Step 6

Multiply $\frac{1}{2} (7 \log_{4} (b))$ .

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Step 6.1

Combine $7$ and $\frac{1}{2}$ .

$\frac{7}{2} \log_{4} (b) + \frac{1}{2} (- \log_{4} (b + 3))$

Step 6.2

Combine $\frac{7}{2}$ and $\log_{4} (b)$ .

$\frac{7 \log_{4} (b)}{2} + \frac{1}{2} (- \log_{4} (b + 3))$

Step 7

Combine $\frac{1}{2}$ and $\log_{4} (b + 3)$ .

$\frac{7 \log_{4} (b)}{2} - \frac{\log_{4} (b + 3)}{2}$

Step 8

Combine the numerators over the common denominator.

$\frac{7 \log_{4} (b) - \log_{4} (b + 3)}{2}$