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

The solution to 1/5 ln(y/x+4)+ln(x) is ln(((y+4x)/x)^{1/5}x)

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

simplify

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

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

Solution steps

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

Apply log rule:

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

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

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

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

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

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What is 1/5 ln(y/x+4)+ln(x) ?
The solution to 1/5 ln(y/x+4)+ln(x) is ln(((y+4x)/x)^{1/5}x)