5^(logx) + x^(log5) = 50 . Find x

Answer 1

5^(logx) + x^(log5) = 50

Note that
log(5^logx)) = logx * log5
log(x^log5)) = log5 * logx

So, the two are equal. That means

2*5^logx = 50
5^logx = 25 = 5^2
logx = 2
x = b^2
where logs are base b.

So, if natural logs, x = e^2
if common logs, x = 10^2 = 100

check (common logs):

5^log100 + 100^log5 = 50
5^2 + 10^log25 = 50
25 + 25 = 50