Precalculus Examples

$\log_{2} ({(x - 1)}^{\frac{3}{2}} {(x + 4)}^{\frac{5}{2}})$

Step 1

Rewrite $\log_{2} ({(x - 1)}^{\frac{3}{2}} {(x + 4)}^{\frac{5}{2}})$ as $\log_{2} ({(x - 1)}^{\frac{3}{2}}) + \log_{2} ({(x + 4)}^{\frac{5}{2}})$ .

$\log_{2} ({(x - 1)}^{\frac{3}{2}}) + \log_{2} ({(x + 4)}^{\frac{5}{2}})$

Step 2

Expand $\log_{2} ({(x - 1)}^{\frac{3}{2}})$ by moving $\frac{3}{2}$ outside the logarithm.

$\frac{3}{2} \log_{2} (x - 1) + \log_{2} ({(x + 4)}^{\frac{5}{2}})$

Step 3

Expand $\log_{2} ({(x + 4)}^{\frac{5}{2}})$ by moving $\frac{5}{2}$ outside the logarithm.

$\frac{3}{2} \log_{2} (x - 1) + \frac{5}{2} \log_{2} (x + 4)$

Step 4

Simplify each term.

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Step 4.1

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

$\frac{3 \log_{2} (x - 1)}{2} + \frac{5}{2} \log_{2} (x + 4)$

Step 4.2

Combine $\frac{5}{2}$ and $\log_{2} (x + 4)$ .

$\frac{3 \log_{2} (x - 1)}{2} + \frac{5 \log_{2} (x + 4)}{2}$

Step 5

Combine the numerators over the common denominator.

$\frac{3 \log_{2} (x - 1) + 5 \log_{2} (x + 4)}{2}$