Precalculus Examples

$\frac{1}{5} \cdot \log (5 x + 1) + \frac{1}{3} \cdot (\log (x - 2) - \log (x^{4} - x^{2} - 5))$

Step 1

Simplify each term.

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Step 1.1

Simplify $\frac{1}{5} \log (5 x + 1)$ by moving $\frac{1}{5}$ inside the logarithm.

$\log ({(5 x + 1)}^{\frac{1}{5}}) + \frac{1}{3} \cdot (\log (x - 2) - \log (x^{4} - x^{2} - 5))$

Step 1.2

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

$\log ({(5 x + 1)}^{\frac{1}{5}}) + \frac{1}{3} \cdot \log (\frac{x - 2}{x^{4} - x^{2} - 5})$

Step 1.3

Simplify $\frac{1}{3} \log (\frac{x - 2}{x^{4} - x^{2} - 5})$ by moving $\frac{1}{3}$ inside the logarithm.

$\log ({(5 x + 1)}^{\frac{1}{5}}) + \log ({(\frac{x - 2}{x^{4} - x^{2} - 5})}^{\frac{1}{3}})$

Step 1.4

Apply the product rule to $\frac{x - 2}{x^{4} - x^{2} - 5}$ .

$\log ({(5 x + 1)}^{\frac{1}{5}}) + \log (\frac{{(x - 2)}^{\frac{1}{3}}}{{(x^{4} - x^{2} - 5)}^{\frac{1}{3}}})$

Step 2

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

$\log ({(5 x + 1)}^{\frac{1}{5}} \frac{{(x - 2)}^{\frac{1}{3}}}{{(x^{4} - x^{2} - 5)}^{\frac{1}{3}}})$

Step 3

Combine ${(5 x + 1)}^{\frac{1}{5}}$ and $\frac{{(x - 2)}^{\frac{1}{3}}}{{(x^{4} - x^{2} - 5)}^{\frac{1}{3}}}$ .

$\log (\frac{{(5 x + 1)}^{\frac{1}{5}} {(x - 2)}^{\frac{1}{3}}}{{(x^{4} - x^{2} - 5)}^{\frac{1}{3}}})$