Precalculus Examples

$\frac{1}{3} \cdot (\log (t) - \log (t + 1)) - \log (t - 1)$

Step 1

Simplify each term.

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Step 1.1

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

$\frac{1}{3} \cdot \log (\frac{t}{t + 1}) - \log (t - 1)$

Step 1.2

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

$\log ({(\frac{t}{t + 1})}^{\frac{1}{3}}) - \log (t - 1)$

Step 1.3

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

$\log (\frac{t^{\frac{1}{3}}}{{(t + 1)}^{\frac{1}{3}}}) - \log (t - 1)$

Step 2

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

$\log (\frac{\frac{t^{\frac{1}{3}}}{{(t + 1)}^{\frac{1}{3}}}}{t - 1})$

Step 3

Multiply the numerator by the reciprocal of the denominator.

$\log (\frac{t^{\frac{1}{3}}}{{(t + 1)}^{\frac{1}{3}}} \cdot \frac{1}{t - 1})$

Step 4

Multiply $\frac{t^{\frac{1}{3}}}{{(t + 1)}^{\frac{1}{3}}}$ by $\frac{1}{t - 1}$ .

$\log (\frac{t^{\frac{1}{3}}}{{(t + 1)}^{\frac{1}{3}} (t - 1)})$