How do you solve #2^x = 3#?
We use logarithms to find that
The inverse of an exponential function is the logarithm. In our question, we have a variable in the exponent of a base which is equal to
We need to do the same thing to both sides of the equation, so it becomes:
But now we need the
To make this work, we need to know how to use a logarithm of a different base than that of the exponent. For this we use the change of base formula
so our equation becomes
Alternate approach to change of base
Another way to look at the change of base is to try using the natural logarithm on the left hand side of the equation, i.e.
We now have to ask ourselves, how do I express
taking the natural log of this equation we also get
Then we could re-write the power of
putting this into the equation above we get
We can substitute for
and finally solving for