Precalculus Examples

$\ln ({(x)}^{3} {(x - 4)}^{2}) (x + 6)$

Step 1

Apply the distributive property.

$\ln (x^{3} {(x - 4)}^{2}) x + \ln (x^{3} {(x - 4)}^{2}) \cdot 6$

Step 2

Multiply $\ln (x^{3} {(x - 4)}^{2}) \cdot 6$ .

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Step 2.1

Reorder $\ln (x^{3} {(x - 4)}^{2})$ and $6$ .

$\ln (x^{3} {(x - 4)}^{2}) x + 6 \cdot \ln (x^{3} {(x - 4)}^{2})$

Step 2.2

Simplify $6 \cdot \ln (x^{3} {(x - 4)}^{2})$ by moving $6$ inside the logarithm.

$\ln (x^{3} {(x - 4)}^{2}) x + \ln ({(x^{3} {(x - 4)}^{2})}^{6})$

Step 3

Simplify each term.

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Step 3.1

Apply the product rule to $x^{3} {(x - 4)}^{2}$ .

$\ln (x^{3} {(x - 4)}^{2}) x + \ln ({(x^{3})}^{6} {({(x - 4)}^{2})}^{6})$

Step 3.2

Multiply the exponents in ${(x^{3})}^{6}$ .

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Step 3.2.1

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

$\ln (x^{3} {(x - 4)}^{2}) x + \ln (x^{3 \cdot 6} {({(x - 4)}^{2})}^{6})$

Step 3.2.2

Multiply $3$ by $6$ .

$\ln (x^{3} {(x - 4)}^{2}) x + \ln (x^{18} {({(x - 4)}^{2})}^{6})$

Step 3.3

Multiply the exponents in ${({(x - 4)}^{2})}^{6}$ .

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Step 3.3.1

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

$\ln (x^{3} {(x - 4)}^{2}) x + \ln (x^{18} {(x - 4)}^{2 \cdot 6})$

Step 3.3.2

Multiply $2$ by $6$ .

$\ln (x^{3} {(x - 4)}^{2}) x + \ln (x^{18} {(x - 4)}^{12})$

Step 4

Reorder factors in $\ln (x^{3} {(x - 4)}^{2}) x + \ln (x^{18} {(x - 4)}^{12})$ .

$x \ln (x^{3} {(x - 4)}^{2}) + \ln (x^{18} {(x - 4)}^{12})$