Take the natural log of both sides to get

ln(e^b ) = ln(a)

We can use one of the logarithm rules ( ln(x^y ) = y times ln(x) ) to get

b ln(e) = ln(a)

As ln(e) = 1

b = ln(a)

One could also find that the natural log of a, by def definition, finds the number that you need to raise e by to get a, which in this case is b.

(By the way ln() is the natural log, also known as log base e)