solving this exponential equation 3^x=2^x-1

Question

solve the exponential equation algebraically

Answer 1

3^x = 2^x-1

log(3^x) = log(2^x-1)

x log(3) = (x-1)log(2)               [log property:  log(a^b) = b log(a)]

x log(3) = x log(2) - log(2)

x log(3) - x log(2)) = -log(2)

x (log(3) - log(2)) = -log(2)

x = -log(2)/(log(3)-log(2)) = log(2)/log(2/3)     [log property: log(a) - log(b) = log(a/b)]

 

Use you calculator if you want a numerical answer.  You can use any base log.