Precalculus Examples

$3 (\ln (x) + 2 \ln (x + 2)) - \frac{1}{2} \cdot \ln (x - 1)$

Step 1

Simplify each term.

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Step 1.1

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

$3 (\ln (x) + \ln ({(x + 2)}^{2})) - \frac{1}{2} \cdot \ln (x - 1)$

Step 1.2

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

$3 \ln (x {(x + 2)}^{2}) - \frac{1}{2} \cdot \ln (x - 1)$

Step 1.3

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

$\ln ({(x {(x + 2)}^{2})}^{3}) - \frac{1}{2} \cdot \ln (x - 1)$

Step 1.4

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

$\ln (x^{3} {({(x + 2)}^{2})}^{3}) - \frac{1}{2} \cdot \ln (x - 1)$

Step 1.5

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

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Step 1.5.1

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

$\ln (x^{3} {(x + 2)}^{2 \cdot 3}) - \frac{1}{2} \cdot \ln (x - 1)$

Step 1.5.2

Multiply $2$ by $3$ .

$\ln (x^{3} {(x + 2)}^{6}) - \frac{1}{2} \cdot \ln (x - 1)$

Step 1.6

Multiply $- \frac{1}{2} \ln (x - 1)$ .

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Step 1.6.1

Reorder $\ln (x - 1)$ and $\frac{1}{2}$ .

$\ln (x^{3} {(x + 2)}^{6}) - (\frac{1}{2} \ln (x - 1))$

Step 1.6.2

Simplify $\frac{1}{2} \ln (x - 1)$ by moving $\frac{1}{2}$ inside the logarithm.

$\ln (x^{3} {(x + 2)}^{6}) - \ln ({(x - 1)}^{\frac{1}{2}})$

Step 2

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

$\ln (\frac{x^{3} {(x + 2)}^{6}}{{(x - 1)}^{\frac{1}{2}}})$