Solving Exponential Equations Examples

Solve e^{2x + 1} = 3

To solve e functions, it's best to do the inverse and take the natural log of each side. Notice how we then could easily solve for x.

Solve lnx = 2.106.

e^lnx = e^2.106
x = e^2.106
x ≈ 8.215

That was easy! Since the e function is the inverse of the natural log function, applying e to both sides of the equation will simplify the left side to just an x. And, bingo. Problem solved.

Solve ln(x + 4) + ln(x – 3) = 3ln2.

First we need to combine left side natural logs with the multiplication property and put the 3 in the exponent on the right side to simplify things.

ln((x + 4)(x – 3)) = ln2³

Now we can e both sides and start solving. This is a tough problem because we will have to FOIL the two binomials to solve for x.

e^{ln((x + 4)(x – 3))} = e^ln23
(x + 4)(x – 3) = 8
x² + 4x – 3x – 12 = 8
x² + 1x – 20 = 0
(x – 5)(x + 4) = 0
x = 5
x = -4

We ended up with a quadratic equation that was factorable. We have two answers x = 5 and x = -4. You thought factoring was a thing of the past, but it will come back to haunt you. Bahahaha!

Solve 

Phew. We combined the e function on the left side of the equation by using the subtraction/division property. Then we took the natural log of each side which helped us get x by itself.