Trying to solve for x
2^(0.5x)= [3^(-0.5x)] + 5/3
I know, the usual, take logs of both sides and bring down the exponent, but I'm having a problem with the RHS of the equation, since I can't take logs seperately, i.e. log(a+b) is NOT equal to log(a)+log(b)
I actually know (graphically) that the answer is x=2, and I know that's correct from substituting. Any ideas on how to approach it (besides trial and error)
If you will work with the formula a little you will be able to write it like this:
as you can observe => =>
The second step is to observe that with x>1 => => in order for equation (1) to be verified.
From the condition => => x<2.5 (3)
From the condition (2) and (3) we have 1 < x< 3 . The only solution for x in this interval in order for and to be rational numbers is x=2;
To demonstrate that this is the only solution it,s easy to study the monotony of the function in interval ( 1,5 ; 2,5)
Have a nice day!