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

The solution to ln(y/x-4)+6(1/5 ln(y/x-1)-1/5 ln(y/x+4))+ln(x) is ln(((y-x)/(y+4x))^{6/5}(y-4x))

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simplify ln(y/x-4)+6(1/5 ln(y/x-1)-1/5 ln(y/x+4))+ln(x)

simplify

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

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

Solution steps

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

Apply log rule:

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

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

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

Simplify

(\frac{y}{x} - 4) x : y - 4 x

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

Apply log rule:

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

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

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

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

Apply log rule:

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

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

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

Simplify

Apply log rule:

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

Apply log rule:

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

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

\frac{3}{8} \cdot ln (\frac{8}{3}) + (\frac{5}{16} \cdot ln (\frac{16}{5})) \cdot 2

ln (\frac{1}{d}) - ln (\frac{1}{y})

ln (\frac{3 x - 4}{2 x + 1})

(\frac{1}{2} lo g_{e} (9 x^{2})) + lo g_{e} (3 x)

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