Precalculus Examples

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

Step 1

Simplify each term.

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Step 1.1

Simplify $4 \ln (x)$ by moving $4$ inside the logarithm.

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

Step 1.2

Simplify $4 \ln (x)$ by moving $4$ inside the logarithm.

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

Step 1.3

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

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

Step 1.4

Multiply $x^{3}$ by $x^{4}$ by adding the exponents.

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Step 1.4.1

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 1.4.2

Add $3$ and $4$ .

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

Step 1.5

Simplify $- 2 \ln (x^{7})$ by moving $2$ inside the logarithm.

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

Step 1.6

Multiply the exponents in ${(x^{7})}^{2}$ .

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Step 1.6.1

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

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

Step 1.6.2

Multiply $7$ by $2$ .

$\ln (x^{4}) - \ln (x^{14})$

Step 2

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

$\ln (\frac{x^{4}}{x^{14}})$

Step 3

Cancel the common factor of $x^{4}$ and $x^{14}$ .

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Step 3.1

Multiply by $1$ .

$\ln (\frac{x^{4} \cdot 1}{x^{14}})$

Step 3.2

Cancel the common factors.

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Step 3.2.1

Factor $x^{4}$ out of $x^{14}$ .

$\ln (\frac{x^{4} \cdot 1}{x^{4} x^{10}})$

Step 3.2.2

Cancel the common factor.

$\ln (\frac{x^{4} \cdot 1}{x^{4} x^{10}})$

Step 3.2.3

Rewrite the expression.

$\ln (\frac{1}{x^{10}})$