What is simplify-1/5 ln|x+1|+1/5 ln|x-4|+C ?

The solution to simplify-1/5 ln|x+1|+1/5 ln|x-4|+C is ln(\sqrt[5]{(|x-4|)/(|x+1|)})+C

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simplify-1/5 ln|x+1|+1/5 ln|x-4|+C

simplify

- \frac{1}{5} ln ∣ x + 1 ∣ + \frac{1}{5} ln ∣ x - 4 ∣ + C

Solution steps

- \frac{1}{5} ln ∣ x + 1 ∣ + \frac{1}{5} ln ∣ x - 4 ∣ + C

Apply log rule:

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

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

= - ln (∣ x + 1 ∣^{\frac{1}{5}}) + \frac{1}{5} ln ∣ x - 4 ∣ + C

Apply log rule:

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

\frac{1}{5} ln ∣ x - 4 ∣ = ln (∣ x - 4 ∣^{\frac{1}{5}})

= - ln (∣ x + 1 ∣^{\frac{1}{5}}) + ln (∣ x - 4 ∣^{\frac{1}{5}}) + C

Apply log rule:

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

ln (∣ x - 4 ∣^{\frac{1}{5}}) - ln (∣ x + 1 ∣^{\frac{1}{5}}) = ln (\frac{∣ x - 4 ∣ ^{\frac{1}{5}}}{∣ x + 1 ∣ ^{\frac{1}{5}}})

= ln (\frac{∣ x - 4 ∣ ^{\frac{1}{5}}}{∣ x + 1 ∣ ^{\frac{1}{5}}}) + C

Combine same powers :

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

\frac{1}{2} ln (\frac{1 + z}{1 - z})

u (ln (u) - ln (x))

lo g_{1.8} (\frac{8.13}{6.09}) + 52.11 - 52.34

\frac{1}{4} ln (x) - \frac{1}{8} ln (x^{2} + 4)

What is simplify-1/5 ln|x+1|+1/5 ln|x-4|+C ?
The solution to simplify-1/5 ln|x+1|+1/5 ln|x-4|+C is ln(\sqrt[5]{(|x-4|)/(|x+1|)})+C