What is 1/2 ln|x|-1/2 ln|x-2| ?

The solution to 1/2 ln|x|-1/2 ln|x-2| is ln(sqrt((|x|)/(|x-2|)))

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simplify 1/2 ln|x|-1/2 ln|x-2|

simplify

\frac{1}{2} ln ∣ x ∣ - \frac{1}{2} ln ∣ x - 2 ∣

ln (\frac{∣ x ∣}{∣ x - 2 ∣})

Solution steps

\frac{1}{2} ln ∣ x ∣ - \frac{1}{2} ln ∣ x - 2 ∣

Apply log rule:

a lo g_{c} (b) = lo g_{c} (b^{a})

\frac{1}{2} ln ∣ x ∣ = ln (∣ x ∣^{\frac{1}{2}})

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

Apply log rule:

a lo g_{c} (b) = lo g_{c} (b^{a})

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

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

Apply log rule:

lo g_{c} (a) - lo g_{c} (b) = lo g_{c} (\frac{a}{b})

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

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

Combine same powers :

= ln (\frac{∣ x ∣}{∣ x - 2 ∣})

- ln (3) + ln (2) + ln (2) - ln (1)

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What is 1/2 ln|x|-1/2 ln|x-2| ?
The solution to 1/2 ln|x|-1/2 ln|x-2| is ln(sqrt((|x|)/(|x-2|)))