1. Solving Exponential Equations Using Logarithms

2. Solving Exponential Functions with Logarithms
There are three main steps to solving exponential functions using logarithms.
Express each side of the equation as a single term.
Take the logarithm of both sides, choosing a convenient base.
Solve the resulting equation.

3. Example
Solve the equation 4 * 2x-3 = 5 * 3x.

4. Solution: Step 1
Conveniently, each side is already expressed as a single term. We can simplify the left side, though, by noticing that 4 = 22, so 4 * 2x-3 is equal to 2x-1.
This simplifies our equation to 2x-1 = 5 * 3x

5. Solution: Step 2 Now we take the logarithm of both sides.
Each side has a different base, so there’s no truly convenient base for us to use.
We’ll use the natural log.
ln(2x-1) = ln(5 * 3x)
Now we can solve the equation using the properties of logarithms.

6. Solution: Step 3 ln(2x-1) = ln(5 * 3x) (x-1)ln(2) = ln(5) + xln(3)
(ln(2) – ln(3))x = ln 5 + ln 2
Using a calculator, we can calculate all of these natural logs.

7. Solution: Part 3 ln 2 = 0.69 ln 3 = 1.10 ln 5 = 1.61
Thus, -0.41x = 2.30
x = -5.6