Precalculus Examples

$\frac{1}{2} \cdot (\log_{9} (x) + \log_{9} (y) - 3 \log_{9} (x + 9))$

Step 1

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

$\frac{1}{2} \cdot (\log_{9} (x y) - 3 \log_{9} (x + 9))$

Step 2

Simplify $- 3 \log_{9} (x + 9)$ by moving $3$ inside the logarithm.

$\frac{1}{2} \cdot (\log_{9} (x y) - \log_{9} ({(x + 9)}^{3}))$

Step 3

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

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

Step 4

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

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

Step 5

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 5.1

Apply the product rule to $\frac{x y}{{(x + 9)}^{3}}$ .

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

Step 5.2

Apply the product rule to $x y$ .

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

Step 6

Multiply the exponents in ${({(x + 9)}^{3})}^{\frac{1}{2}}$ .

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Step 6.1

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

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

Step 6.2

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

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