Precalculus Examples

$6 \log_{5} (w) + \frac{\log_{5} (u)}{2} + \frac{\log_{5} (v)}{2}$

Step 1

Combine the numerators over the common denominator.

$6 \log_{5} (w) + \frac{\log_{5} (u) + \log_{5} (v)}{2}$

Step 2

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

$6 \log_{5} (w) + \frac{\log_{5} (u v)}{2}$

Step 3

Simplify each term.

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Step 3.1

Simplify $6 \log_{5} (w)$ by moving $6$ inside the logarithm.

$\log_{5} (w^{6}) + \frac{\log_{5} (u v)}{2}$

Step 3.2

Rewrite $\frac{\log_{5} (u v)}{2}$ as $\frac{1}{2} \log_{5} (u v)$ .

$\log_{5} (w^{6}) + \frac{1}{2} \log_{5} (u v)$

Step 3.3

Simplify $\frac{1}{2} \log_{5} (u v)$ by moving $\frac{1}{2}$ inside the logarithm.

$\log_{5} (w^{6}) + \log_{5} ({(u v)}^{\frac{1}{2}})$

Step 3.4

Apply the product rule to $u v$ .

$\log_{5} (w^{6}) + \log_{5} (u^{\frac{1}{2}} v^{\frac{1}{2}})$

Step 4

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

$\log_{5} (w^{6} u^{\frac{1}{2}} v^{\frac{1}{2}})$