# How do you solve #2^x = 3#?

##### 1 Answer

We use logarithms to find that

#### Explanation:

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

which gives

We can substitute for

#ln(2)*x=ln(3)

and finally solving for