What is 1/(2(x-1))+5/8 ln|x-1|-5/8 ln|x+3| ?

The solution to 1/(2(x-1))+5/8 ln|x-1|-5/8 ln|x+3| is 1/(2(x-1))+ln(\sqrt[8]{(|x-1|^5)/(|x+3|^5)})

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simplify 1/(2(x-1))+5/8 ln|x-1|-5/8 ln|x+3|

simplify

\frac{1}{2 ( x - 1 )} + \frac{5}{8} ln ∣ x - 1 ∣ - \frac{5}{8} ln ∣ x + 3 ∣

Solution steps

\frac{1}{2 ( x - 1 )} + \frac{5}{8} ln ∣ x - 1 ∣ - \frac{5}{8} ln ∣ x + 3 ∣

Apply log rule:

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

\frac{5}{8} ln ∣ x - 1 ∣ = ln (∣ x - 1 ∣^{\frac{5}{8}})

= \frac{1}{2 ( x - 1 )} + ln (∣ x - 1 ∣^{\frac{5}{8}}) - \frac{5}{8} ln ∣ x + 3 ∣

Apply log rule:

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

\frac{5}{8} ln ∣ x + 3 ∣ = ln (∣ x + 3 ∣^{\frac{5}{8}})

= \frac{1}{2 ( x - 1 )} + ln (∣ x - 1 ∣^{\frac{5}{8}}) - ln (∣ x + 3 ∣^{\frac{5}{8}})

Apply log rule:

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

ln (∣ x - 1 ∣^{\frac{5}{8}}) - ln (∣ x + 3 ∣^{\frac{5}{8}}) = ln (\frac{∣ x - 1 ∣ ^{\frac{5}{8}}}{∣ x + 3 ∣ ^{\frac{5}{8}}})

= \frac{1}{2 ( x - 1 )} + ln (\frac{∣ x - 1 ∣ ^{\frac{5}{8}}}{∣ x + 3 ∣ ^{\frac{5}{8}}})

Combine same powers :

6 ln (\frac{3}{2}) - 2

ln \frac{5}{4} + 2 ln ∣ - 1 ∣ - 2 ln ∣ - 1 ∣ - 2 ln ∣ - 2 ∣

- ln ∣ 1 ∣ + \frac{1}{e ^{2}} ln e^{2} + 1 - ln \frac{1}{e ^{2}} + 1

lo g_{e} (\frac{1 - \frac{i}{2}}{- 1 - \frac{i}{2}})

What is 1/(2(x-1))+5/8 ln|x-1|-5/8 ln|x+3| ?
The solution to 1/(2(x-1))+5/8 ln|x-1|-5/8 ln|x+3| is 1/(2(x-1))+ln(\sqrt[8]{(|x-1|^5)/(|x+3|^5)})