Precalculus Examples

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

Step 1

Simplify each term.

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

Apply the distributive property.

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

Step 1.4

Multiply ${(x - 1)}^{2}$ by $1$ .

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

Step 1.5

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

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

Step 1.6

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

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

Step 2

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

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

Step 3

Reorder factors in $\ln ({(x {(x - 1)}^{2} + {(x - 1)}^{2})}^{\frac{1}{2}} x^{3})$ .

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