Precalculus Examples

$2 \ln (x - 1) - \frac{1}{2} \cdot \ln (x) - \ln (x - 1)$

Step 1

Simplify each term.

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Step 1.1

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

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

Step 1.2

Multiply $- \frac{1}{2} \ln (x)$ .

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Step 1.2.1

Reorder $\ln (x)$ and $\frac{1}{2}$ .

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

Step 1.2.2

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

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

Step 2

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

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

Step 3

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

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

Step 4

Multiply the numerator by the reciprocal of the denominator.

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

Step 5

Cancel the common factor of $x - 1$ .

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Step 5.1

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

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

Step 5.2

Cancel the common factor.

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

Step 5.3

Rewrite the expression.

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