Precalculus Examples

$\log_{9} (z) + \frac{\log_{9} (x)}{3} + \frac{\log_{9} (x)}{3}$

Step 1

Combine the numerators over the common denominator.

$\log_{9} (z) + \frac{\log_{9} (x) + \log_{9} (x)}{3}$

Step 2

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

$\log_{9} (z) + \frac{\log_{9} (x \cdot x)}{3}$

Step 3

Multiply $x$ by $x$ .

$\log_{9} (z) + \frac{\log_{9} (x^{2})}{3}$

Step 4

Simplify each term.

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Step 4.1

Rewrite $\frac{\log_{9} (x^{2})}{3}$ as $\frac{1}{3} \log_{9} (x^{2})$ .

$\log_{9} (z) + \frac{1}{3} \log_{9} (x^{2})$

Step 4.2

Simplify $\frac{1}{3} \log_{9} (x^{2})$ by moving $\frac{1}{3}$ inside the logarithm.

$\log_{9} (z) + \log_{9} ({(x^{2})}^{\frac{1}{3}})$

Step 4.3

Multiply the exponents in ${(x^{2})}^{\frac{1}{3}}$ .

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Step 4.3.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\log_{9} (z) + \log_{9} (x^{2 (\frac{1}{3})})$

Step 4.3.2

Combine $2$ and $\frac{1}{3}$ .

$\log_{9} (z) + \log_{9} (x^{\frac{2}{3}})$

Step 5

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

$\log_{9} (z x^{\frac{2}{3}})$