Precalculus Examples

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

Step 1

Apply the product rule to $x (x - 2)$ .

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

Step 2

Rewrite $\frac{x - 1}{x^{2} {(x - 2)}^{2}}$ as ${(\frac{1}{x (x - 2)})}^{2} (x - 1)$ .

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Step 2.1

Factor the perfect power $1^{2}$ out of $x - 1$ .

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

Step 2.2

Factor the perfect power ${(x (x - 2))}^{2}$ out of $x^{2} {(x - 2)}^{2}$ .

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

Step 2.3

Rearrange the fraction $\frac{1^{2} (x - 1)}{{(x (x - 2))}^{2} \cdot 1}$ .

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

Step 3

Pull terms out from under the radical.

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

Step 4

Combine $\frac{1}{x (x - 2)}$ and $\sqrt{x - 1}$ .

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