Precalculus Examples

$\log_{b} (x^{\frac{1}{2}} y^{\frac{1}{3}})$

Step 1

Rewrite $\log_{b} (x^{\frac{1}{2}} y^{\frac{1}{3}})$ as $\log_{b} (x^{\frac{1}{2}}) + \log_{b} (y^{\frac{1}{3}})$ .

$\log_{b} (x^{\frac{1}{2}}) + \log_{b} (y^{\frac{1}{3}})$

Step 2

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

$\frac{1}{2} \log_{b} (x) + \log_{b} (y^{\frac{1}{3}})$

Step 3

Expand $\log_{b} (y^{\frac{1}{3}})$ by moving $\frac{1}{3}$ outside the logarithm.

$\frac{1}{2} \log_{b} (x) + \frac{1}{3} \log_{b} (y)$

Step 4

Simplify each term.

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Step 4.1

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

$\frac{\log_{b} (x)}{2} + \frac{1}{3} \log_{b} (y)$

Step 4.2

Combine $\frac{1}{3}$ and $\log_{b} (y)$ .

$\frac{\log_{b} (x)}{2} + \frac{\log_{b} (y)}{3}$