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

The solution to ln(y/x)+ln(1/(x+y))-ln(x/6) is ln((6y)/(x^2(x+y)))

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

simplify

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

ln (\frac{6 y}{x ^{2} ( x + y )})

Solution steps

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

Apply log rule:

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

ln (\frac{y}{x}) + ln (\frac{1}{x + y}) = ln (\frac{y}{x} \cdot \frac{1}{y + x})

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

Simplify

\frac{y}{x} \cdot \frac{1}{x + y} : \frac{y}{x ( x + y )}

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

Apply log rule:

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

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

= ln (\frac{\frac{y}{x ( x + y )}}{\frac{x}{6}})

Simplify

\frac{\frac{y}{x ( x + y )}}{\frac{x}{6}} : \frac{6 y}{x ^{2} ( x + y )}

= ln (\frac{6 y}{x ^{2} ( x + y )})

- lo g_{10} (4.6 \cdot 1 0^{- 3})

- ln (cos (x)) + ln (1 + sin (x))

ln (a b^{n})

ln (1 - x) + ln (1 + x)

ln (\frac{2 y ^{2}}{z ^{3}})

What is ln(y/x)+ln(1/(x+y))-ln(x/6) ?
The solution to ln(y/x)+ln(1/(x+y))-ln(x/6) is ln((6y)/(x^2(x+y)))