Precalculus Examples

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

Step 1

Simplify each term.

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Step 1.1

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

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

Step 1.2

Multiply $- \frac{1}{2} \log (x^{2} + 1)$ .

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Step 1.2.1

Reorder $\log (x^{2} + 1)$ and $\frac{1}{2}$ .

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

Step 1.2.2

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

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

Step 2

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

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

Step 3

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

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