Precalculus Examples

$4 (\ln (x) + 3 \ln (x + 2)) - 3 \ln (2)$

Step 1

Simplify each term.

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Step 1.1

Simplify $3 \ln (x + 2)$ by moving $3$ inside the logarithm.

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

Step 1.2

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

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

Step 1.3

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

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

Step 1.4

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

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

Step 1.5

Multiply the exponents in ${({(x + 2)}^{3})}^{4}$ .

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Step 1.5.1

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

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

Step 1.5.2

Multiply $3$ by $4$ .

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

Step 1.6

Simplify $- 3 \ln (2)$ by moving $3$ inside the logarithm.

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

Step 1.7

Raise $2$ to the power of $3$ .

$\ln (x^{4} {(x + 2)}^{12}) - \ln (8)$

Step 2

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

$\ln (\frac{x^{4} {(x + 2)}^{12}}{8})$