log-x-y-pi-log-y-x-e-a-1-log-y-x-pi-1-1-log-x-y-e-1-b-x-a-b-2e-y-a-b-2pi-

Question Number 647 by 123456 last updated on 18/Feb/15

\log_{x} (y^{π}) + \log_{y} (x^{e}) = a

\frac{1}{\log_{y} (x^{π^{- 1}})} - \frac{1}{\log_{x} (y^{e^{- 1}})} = b

\frac{x^{a + b + 2 e}}{y^{a - b + 2 π}} = ?

Commented by 123456 last updated on 17/Feb/15

\log_{x} (y^{π}) + \log_{y} (x^{e}) = a

π \log_{x} y + e \log_{y} x = a

\log_{y} x = \frac{\log_{x} x}{\log_{x} y} = \frac{1}{\log_{x} y}

\frac{1}{\log_{y} (x^{π^{- 1}})} - \frac{1}{\log_{x} (y^{e^{- 1}})} = b

\frac{π}{\log_{y} x} - \frac{e}{\log_{x} y} = b

π \log_{x} y - e \log_{y} x = b

a + b = (π \log_{x} y + e \log_{y} x) + (π \log_{x} y - e \log_{y} x) = 2 π \log_{x} y

a - b = (π \log_{x} y + e \log_{y} x) - (π \log_{x} y - e \log_{y} x) = 2 e \log_{y} x

Answered by prakash jain last updated on 19/Feb/15

\frac{x^{a + b + 2 e}}{y^{a - b + 2 π}} = \frac{x^{2 π \log_{x} y + 2 e}}{y^{2 e \log_{y} x + 2 π}} = \frac{y^{2 π} x^{2 e}}{x^{2 e} y^{2 π}} = 1