Solve for x,y: (3x)^(log3) = (4y)^(log4) and 4^(log x) = 3^(log y) then how do I solve for x?

Josiah Owens

Josiah Owens

Answered question

2022-10-23

Solve for x , y: ( 3 x ) log 3 = ( 4 y ) log 4 and 4 log x = 3 log y
then how do I solve for x?
I tried taking log on both sides but after few steps I got stuck

Answer & Explanation

megagoalai

megagoalai

Beginner2022-10-24Added 22 answers

When you take the logarithm on each side of each equation, the two equations become
( log 3 ) ( log 3 + log x ) = ( log 4 ) ( log 4 + log y ) and ( log x ) ( log 4 ) = ( log y ) ( log 3 )
Solving the second equation for log y gives
log y = log 4 log 3 log x
Plugging this into the first equation gives
( log 3 ) ( log 3 + log x ) = ( log 4 ) ( log 4 + log 4 log 3 log x )
which, on moving stuff around, becomes
( log 3 ( log 4 ) 2 log 3 ) log x = ( log 4 ) 2 ( log 3 ) 2
It should be easy from here to see that
log x log 3 = 1
which implies log x = log 3 = log ( 1 / 3 ), so that x = 1 / 3
Sonia Elliott

Sonia Elliott

Beginner2022-10-25Added 4 answers

I believe a hint is in order
4 log x = 3 log y log 4 log x = log 3 log y y = 10 log 4 log 3 log x = x log 4 log 3
Insert into the other one. Re-arrange the first equation to obtain a similar exponent which should be
x λ ( log 2 4 log 2 3 ) = 10 log 2 4 log 2 3
it should become clear.

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