Precalculus Examples

$4 \ln (x^{7} y) - 2 \ln (x y^{6})$

Step 1

Simplify each term.

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Step 1.1

Simplify $4 \ln (x^{7} y)$ by moving $4$ inside the logarithm.

$\ln ({(x^{7} y)}^{4}) - 2 \ln (x y^{6})$

Step 1.2

Apply the product rule to $x^{7} y$ .

$\ln ({(x^{7})}^{4} y^{4}) - 2 \ln (x y^{6})$

Step 1.3

Multiply the exponents in ${(x^{7})}^{4}$ .

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Step 1.3.1

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

$\ln (x^{7 \cdot 4} y^{4}) - 2 \ln (x y^{6})$

Step 1.3.2

Multiply $7$ by $4$ .

$\ln (x^{28} y^{4}) - 2 \ln (x y^{6})$

Step 1.4

Simplify $- 2 \ln (x y^{6})$ by moving $2$ inside the logarithm.

$\ln (x^{28} y^{4}) - \ln ({(x y^{6})}^{2})$

Step 1.5

Apply the product rule to $x y^{6}$ .

$\ln (x^{28} y^{4}) - \ln (x^{2} {(y^{6})}^{2})$

Step 1.6

Multiply the exponents in ${(y^{6})}^{2}$ .

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Step 1.6.1

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

$\ln (x^{28} y^{4}) - \ln (x^{2} y^{6 \cdot 2})$

Step 1.6.2

Multiply $6$ by $2$ .

$\ln (x^{28} y^{4}) - \ln (x^{2} y^{12})$

Step 2

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

$\ln (\frac{x^{28} y^{4}}{x^{2} y^{12}})$

Step 3

Cancel the common factor of $x^{28}$ and $x^{2}$ .

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Step 3.1

Factor $x^{2}$ out of $x^{28} y^{4}$ .

$\ln (\frac{x^{2} (x^{26} y^{4})}{x^{2} y^{12}})$

Step 3.2

Cancel the common factors.

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Step 3.2.1

Factor $x^{2}$ out of $x^{2} y^{12}$ .

$\ln (\frac{x^{2} (x^{26} y^{4})}{x^{2} (y^{12})})$

Step 3.2.2

Cancel the common factor.

$\ln (\frac{x^{2} (x^{26} y^{4})}{x^{2} y^{12}})$

Step 3.2.3

Rewrite the expression.

$\ln (\frac{x^{26} y^{4}}{y^{12}})$

Step 4

Cancel the common factor of $y^{4}$ and $y^{12}$ .

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Step 4.1

Factor $y^{4}$ out of $x^{26} y^{4}$ .

$\ln (\frac{y^{4} x^{26}}{y^{12}})$

Step 4.2

Cancel the common factors.

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Step 4.2.1

Factor $y^{4}$ out of $y^{12}$ .

$\ln (\frac{y^{4} x^{26}}{y^{4} y^{8}})$

Step 4.2.2

Cancel the common factor.

$\ln (\frac{y^{4} x^{26}}{y^{4} y^{8}})$

Step 4.2.3

Rewrite the expression.

$\ln (\frac{x^{26}}{y^{8}})$