Precalculus Examples

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

Step 1

Simplify each term.

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Step 1.1

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

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

Step 1.2

Simplify $- 4 \ln (\frac{1}{2})$ by moving $4$ inside the logarithm.

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

Step 1.3

Apply the product rule to $\frac{1}{2}$ .

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

Step 1.4

One to any power is one.

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

Step 1.5

Raise $2$ to the power of $4$ .

$\ln ({(x^{2} + 1)}^{\frac{1}{2}}) - \ln (\frac{1}{16}) - \frac{1}{2} \cdot (\ln (x - 4) \ln (x))$

Step 1.6

Multiply $- \frac{1}{2} (\ln (x - 4) \ln (x))$ .

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Step 1.6.1

Reorder $\ln (x - 4)$ and $\frac{1}{2}$ .

$\ln ({(x^{2} + 1)}^{\frac{1}{2}}) - \ln (\frac{1}{16}) - (\frac{1}{2} \ln (x - 4)) \ln (x)$

Step 1.6.2

Simplify $\frac{1}{2} \ln (x - 4)$ by moving $\frac{1}{2}$ inside the logarithm.

$\ln ({(x^{2} + 1)}^{\frac{1}{2}}) - \ln (\frac{1}{16}) - \ln ({(x - 4)}^{\frac{1}{2}}) \ln (x)$

Step 2

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

$\ln (\frac{{(x^{2} + 1)}^{\frac{1}{2}}}{\frac{1}{16}}) - \ln ({(x - 4)}^{\frac{1}{2}}) \ln (x)$

Step 3

Simplify each term.

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Step 3.1

Multiply the numerator by the reciprocal of the denominator.

$\ln ({(x^{2} + 1)}^{\frac{1}{2}} \cdot 16) - \ln ({(x - 4)}^{\frac{1}{2}}) \ln (x)$

Step 3.2

Move $16$ to the left of ${(x^{2} + 1)}^{\frac{1}{2}}$ .

$\ln (16 {(x^{2} + 1)}^{\frac{1}{2}}) - \ln ({(x - 4)}^{\frac{1}{2}}) \ln (x)$