Precalculus Examples

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

Step 1

Simplify each term.

Tap for more steps...

Step 1.1

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

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

Step 1.2

Simplify $- 3 \log (x - 1)$ by moving $3$ inside the logarithm.

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

Step 2

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

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

Step 3

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

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

Step 4

Multiply the numerator by the reciprocal of the denominator.

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

Step 5

Combine.

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

Step 6

Cancel the common factor of ${(x - 1)}^{2}$ and ${(x - 1)}^{3}$ .

Tap for more steps...

Step 6.1

Factor ${(x - 1)}^{2}$ out of ${(x - 1)}^{2} \cdot 1$ .

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

Step 6.2

Cancel the common factors.

Tap for more steps...

Step 6.2.1

Factor ${(x - 1)}^{2}$ out of $x^{3} {(x - 1)}^{3}$ .

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

Step 6.2.2

Cancel the common factor.

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

Step 6.2.3

Rewrite the expression.

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