Precalculus Examples

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

Step 1

Simplify each term.

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Step 1.1

Simplify $4 \log (x)$ by moving $4$ inside the logarithm.

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

Step 1.2

Multiply $- \frac{1}{3} \log (x^{2} + 1)$ .

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Step 1.2.1

Reorder $\log (x^{2} + 1)$ and $\frac{1}{3}$ .

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

Step 1.2.2

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

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

Step 1.3

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

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

Step 2

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

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

Step 3

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

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

Step 4

Combine $\frac{x^{4}}{{(x^{2} + 1)}^{\frac{1}{3}}}$ and ${(x - 1)}^{2}$ .

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