Precalculus Examples

$\ln (x^{2} - 2 \ln (\sqrt{x}))$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

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

Step 2

Expand $\ln (x^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

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

Step 3

Combine $\frac{1}{2}$ and $- 2$ .

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

Step 4

Reduce the expression $\frac{- 2}{2}$ by cancelling the common factors.

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Step 4.1

Factor $2$ out of $- 2$ .

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

Step 4.2

Factor $2$ out of $2$ .

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

Step 4.3

Cancel the common factor.

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

Step 4.4

Rewrite the expression.

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

Step 5

Divide $- 1$ by $1$ .

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

Step 6

Rewrite $- 1 \ln (x)$ as $- \ln (x)$ .

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