Precalculus Examples

$\frac{1}{2} \ln (x^{3}) + 3 \ln (x - 4)$

Step 1

Simplify each term.

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Step 1.1

Simplify $\frac{1}{2} \ln (x^{3})$ by moving $\frac{1}{2}$ inside the logarithm.

$\ln ({(x^{3})}^{\frac{1}{2}}) + 3 \ln (x - 4)$

Step 1.2

Multiply the exponents in ${(x^{3})}^{\frac{1}{2}}$ .

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Step 1.2.1

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

$\ln (x^{3 (\frac{1}{2})}) + 3 \ln (x - 4)$

Step 1.2.2

Combine $3$ and $\frac{1}{2}$ .

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

Step 1.3

Simplify $3 \ln (x - 4)$ by moving $3$ inside the logarithm.

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

Step 2

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

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