What is 1/4 ln(y/x-1)-5/4 ln(y/x+3)-ln(x) ?

The solution to 1/4 ln(y/x-1)-5/4 ln(y/x+3)-ln(x) is ln(\sqrt[4]{(y-x)/((y+3x)^5)})

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simplify 1/4 ln(y/x-1)-5/4 ln(y/x+3)-ln(x)

simplify

\frac{1}{4} ln (\frac{y}{x} - 1) - \frac{5}{4} ln (\frac{y}{x} + 3) - ln (x)

Solution steps

\frac{1}{4} ln (\frac{y}{x} - 1) - \frac{5}{4} ln (\frac{y}{x} + 3) - ln (x)

Apply log rule:

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

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

= ln ((\frac{y}{x} - 1)^{\frac{1}{4}}) - \frac{5}{4} ln (\frac{y}{x} + 3) - ln (x)

Apply log rule:

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

\frac{5}{4} ln (\frac{y}{x} + 3) = ln ((\frac{y}{x} + 3)^{\frac{5}{4}})

= ln ((\frac{y}{x} - 1)^{\frac{1}{4}}) - ln ((\frac{y}{x} + 3)^{\frac{5}{4}}) - 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{1}{4}}) - ln ((\frac{y}{x} + 3)^{\frac{5}{4}}) = ln \frac{( \frac{y}{x} - 1 ) ^{\frac{1}{4}}}{( \frac{y}{x} + 3 ) ^{\frac{5}{4}}}

= ln \frac{( \frac{y}{x} - 1 ) ^{\frac{1}{4}}}{( \frac{y}{x} + 3 ) ^{\frac{5}{4}}} - ln (x)

Simplify

Apply log rule:

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

Simplify

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ln (2 y)

What is 1/4 ln(y/x-1)-5/4 ln(y/x+3)-ln(x) ?
The solution to 1/4 ln(y/x-1)-5/4 ln(y/x+3)-ln(x) is ln(\sqrt[4]{(y-x)/((y+3x)^5)})