Precalculus Examples

$9 \log ({(x)}^{2}) - 12 \log (x) \log (y) + 4 \log ({(y)}^{2})$

Step 1

Simplify each term.

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Step 1.1

Simplify $9 \log (x^{2})$ by moving $9$ inside the logarithm.

$\log ({(x^{2})}^{9}) - 12 \log (x) \log (y) + 4 \log ({(y)}^{2})$

Step 1.2

Multiply the exponents in ${(x^{2})}^{9}$ .

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Step 1.2.1

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

$\log (x^{2 \cdot 9}) - 12 \log (x) \log (y) + 4 \log ({(y)}^{2})$

Step 1.2.2

Multiply $2$ by $9$ .

$\log (x^{18}) - 12 \log (x) \log (y) + 4 \log ({(y)}^{2})$

Step 1.3

Simplify $- 12 \log (x)$ by moving $12$ inside the logarithm.

$\log (x^{18}) - \log (x^{12}) \log (y) + 4 \log ({(y)}^{2})$

Step 1.4

Simplify $4 \log (y^{2})$ by moving $4$ inside the logarithm.

$\log (x^{18}) - \log (x^{12}) \log (y) + \log ({(y^{2})}^{4})$

Step 1.5

Multiply the exponents in ${(y^{2})}^{4}$ .

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Step 1.5.1

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

$\log (x^{18}) - \log (x^{12}) \log (y) + \log (y^{2 \cdot 4})$

Step 1.5.2

Multiply $2$ by $4$ .

$\log (x^{18}) - \log (x^{12}) \log (y) + \log (y^{8})$

Step 2

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

$- \log (x^{12}) \log (y) + \log (x^{18} y^{8})$