Log is an Injective function?

(1) Algebraic method:
log x1 = log x2
log x1 - log x2 = 0
log (x1/x2) = 0
x1/x2 = 1
x1 = x2
(2) Calculus method:
d(log x)/dx = d[(ln x)/(ln 10)]/dx
= 1/(x ln 10)
So in the domain of log, i.e. all positive real values of x, d(log x)/dx > 0 and hence log x is strictly increasing.
In conclusion, log x is injective.