What is 6/5 ln| y/x-1|-1/5 ln| y/x+4|+ln|x| ?

The solution to 6/5 ln| y/x-1|-1/5 ln| y/x+4|+ln|x| is ln(\sqrt[5]{((y-x)^6)/(|y+4x|)})

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simplify 6/5 ln| y/x-1|-1/5 ln| y/x+4|+ln|x|

simplify

\frac{6}{5} ln \frac{y}{x} - 1 - \frac{1}{5} ln \frac{y}{x} + 4 + ln ∣ x ∣

Solution steps

\frac{6}{5} ln \frac{y}{x} - 1 - \frac{1}{5} ln \frac{y}{x} + 4 + ln ∣ x ∣

Apply log rule:

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

\frac{6}{5} ln \frac{y}{x} - 1 = ln (\frac{y}{x} - 1^{\frac{6}{5}})

= ln (\frac{y}{x} - 1^{\frac{6}{5}}) - \frac{1}{5} ln \frac{y}{x} + 4 + ln ∣ x ∣

Apply log rule:

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

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

= ln (\frac{y}{x} - 1^{\frac{6}{5}}) - ln (\frac{y}{x} + 4^{\frac{1}{5}}) + ln ∣ x ∣

Apply log rule:

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

ln (\frac{y}{x} - 1^{\frac{6}{5}}) - ln (\frac{y}{x} + 4^{\frac{1}{5}}) = ln \frac{\frac{y}{x} - 1 ^{\frac{6}{5}}}{\frac{y}{x} + 4 ^{\frac{1}{5}}}

= ln \frac{\frac{y}{x} - 1 ^{\frac{6}{5}}}{\frac{y}{x} + 4 ^{\frac{1}{5}}} + ln ∣ x ∣

Simplify

\frac{\frac{y}{x} - 1 ^{\frac{6}{5}}}{\frac{y}{x} + 4 ^{\frac{1}{5}}} : \frac{∣ y - x ∣ ^{\frac{6}{5}} ∣ y + 4 x ∣ ^{\frac{4}{5}}}{∣ x ∣ ∣ y + 4 x ∣}

= ln (\frac{∣ y - x ∣ ^{\frac{6}{5}} ∣ y + 4 x ∣ ^{\frac{4}{5}}}{∣ x ∣ ∣ y + 4 x ∣}) + ln ∣ x ∣

Apply log rule:

lo g_{c} (a) + lo g_{c} (b) = lo g_{c} (ab)

ln (\frac{∣ y - x ∣ ^{\frac{6}{5}} ∣ y + 4 x ∣ ^{\frac{4}{5}}}{∣ x ∣ ∣ y + 4 x ∣}) + ln ∣ x ∣ = ln (\frac{∣ y - x ∣ ^{\frac{6}{5}} ∣ y + 4 x ∣ ^{\frac{4}{5}}}{∣ x ∣ ∣ y + 4 x ∣} ∣ x ∣)

= ln (\frac{∣ y - x ∣ ^{\frac{6}{5}} ∣ y + 4 x ∣ ^{\frac{4}{5}}}{∣ x ∣ ∣ y + 4 x ∣} ∣ x ∣)

Simplify

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ln (\frac{( - 2 i ) ^{2}}{1 + i})

What is 6/5 ln| y/x-1|-1/5 ln| y/x+4|+ln|x| ?
The solution to 6/5 ln| y/x-1|-1/5 ln| y/x+4|+ln|x| is ln(\sqrt[5]{((y-x)^6)/(|y+4x|)})