Question

Solve for x. 6^(x/2)=5^(1-x)

Answer 1

Hello from MrBillDoesMath!

Answer:

x = log5 / ( (1/2)log6 + log5)

    approximately  0.642

Discussion:

Take the logarithm of both sides ( to "pull" the exponents down so we can work on them!)

6^(x/2)) = 5^(1-x)

log ( 6^(x/2)) = log (5^(1-x))      =>   using log (a^n) = n log a

(x/2) log6 = (1-x) log5               =>   add x log5 to both sides

(x/2) log6 + xlog5 =  log5 - xlog5 + xlog5   =>

(x/2) log6 + xlog5 = log5          => factor x from the lhs

x (  (1/2) log6 + log5) = log5     =>

x = log5 / ( (1/2)log6 + log5)    

The above can be further simplified but that's as far as I want to take it.The value of x is approx equal to  0.642

Thank you,

MrB