Precalculus Examples

$\log_{9} (z) + \frac{\log_{9} (x)}{2} + \frac{\log_{9} (y)}{2}$

Step 1

Combine the numerators over the common denominator.

$\log_{9} (z) + \frac{\log_{9} (x) + \log_{9} (y)}{2}$

Step 2

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

$\log_{9} (z) + \frac{\log_{9} (x y)}{2}$

Step 3

Simplify each term.

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Step 3.1

Rewrite $\frac{\log_{9} (x y)}{2}$ as $\frac{1}{2} \log_{9} (x y)$ .

$\log_{9} (z) + \frac{1}{2} \log_{9} (x y)$

Step 3.2

Simplify $\frac{1}{2} \log_{9} (x y)$ by moving $\frac{1}{2}$ inside the logarithm.

$\log_{9} (z) + \log_{9} ({(x y)}^{\frac{1}{2}})$

Step 3.3

Apply the product rule to $x y$ .

$\log_{9} (z) + \log_{9} (x^{\frac{1}{2}} y^{\frac{1}{2}})$

Step 4

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

$\log_{9} (z x^{\frac{1}{2}} y^{\frac{1}{2}})$