What is simplify-1/4 ln|x+2|+1/4 ln|x-2| ?

The solution to simplify-1/4 ln|x+2|+1/4 ln|x-2| is ln(\sqrt[4]{(|x-2|)/(|x+2|)})

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simplify-1/4 ln|x+2|+1/4 ln|x-2|

simplify

- \frac{1}{4} ln ∣ x + 2 ∣ + \frac{1}{4} ln ∣ x - 2 ∣

Solution steps

- \frac{1}{4} ln ∣ x + 2 ∣ + \frac{1}{4} ln ∣ x - 2 ∣

Apply log rule:

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

\frac{1}{4} ln ∣ x + 2 ∣ = ln (∣ x + 2 ∣^{\frac{1}{4}})

= - ln (∣ x + 2 ∣^{\frac{1}{4}}) + \frac{1}{4} ln ∣ x - 2 ∣

Apply log rule:

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

\frac{1}{4} ln ∣ x - 2 ∣ = ln (∣ x - 2 ∣^{\frac{1}{4}})

= - ln (∣ x + 2 ∣^{\frac{1}{4}}) + ln (∣ x - 2 ∣^{\frac{1}{4}})

Apply log rule:

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

ln (∣ x - 2 ∣^{\frac{1}{4}}) - ln (∣ x + 2 ∣^{\frac{1}{4}}) = ln (\frac{∣ x - 2 ∣ ^{\frac{1}{4}}}{∣ x + 2 ∣ ^{\frac{1}{4}}})

= ln (\frac{∣ x - 2 ∣ ^{\frac{1}{4}}}{∣ x + 2 ∣ ^{\frac{1}{4}}})

Combine same powers :

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

\frac{1}{3} lo g_{3} (7) - lo g_{3} (2)

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

ln (2 e^{- 4 x^{2}})

lo g_{7} (e) - 8 lo g_{7} (y) - \frac{1}{8} lo g_{7} (89)

What is simplify-1/4 ln|x+2|+1/4 ln|x-2| ?
The solution to simplify-1/4 ln|x+2|+1/4 ln|x-2| is ln(\sqrt[4]{(|x-2|)/(|x+2|)})