solve the exponential function for x by using common logarithms.
5^x=50^x+1
I will assume you mean
5^x = 50^(x+1)
take logs of both sides
log (5^x) = log (50^(x+1))
use rules of logs
x log5 = (x+1)log50
x log5 = xlog50 + log50
xlog5 - xlog50 = log50
x(log5 - log50) = log50
x = log50/(log5 - log50)
= appr -1.699
check:
5^-1.699 = .06493..
50^-.699 = .064926..
not bad