I. Solving Exponential Equations
A. Procedure
1. Isolate the exponential.
2. Take the log of both sides.
a. If the base of the exp onential is e , use ln x.
b. Otherwise, use log x.
c. Remember that 
.
d. Remember the Power Property , that 
.
3. Solve for the variable .
4. Round your answer.
5. Remember; don't round off until the bitter end!
B. Examples - Solve each of the fol lowing .
First, we notice that the exponential is e x, so we need to get it all by itself.
So divide
both sides by 4 to get:
Now take the natural log of both sides to get:
But what is the base of ln x? So we have:
Answer: x = 3.1246
The exponential is isolated, so we can take the common log of both sides:
Now we use the Power Property to get:
Believe it or not, but log 8 is just a number. It is an ugly number, but it's
still just a
number. So to solve this, we need to divide both sides by log 8
(just like we would if
this were 8x = 4).
Answer:
3. Now you try one: 
Answer: x = 0
On this one, both exponentials are isolated, so we can take the common log of
both
sides to get:
Using the Power Property, we get:
Again, log 6 is just a number, ugly, but a number. So we distribute to get :
Now, we need to get the terms with "x" in them together on
one side. So let' s subtract
from both sides to get:
Now we factor out the GCF on the right to get:
So now we can divide both sides by the coefficient of x to get :
Answer: -13.2571
5. If $1,000 is deposited today at 5% compounded continuously , how many years
will it take
to double the investment? Round your answer to the nearest tenth of
a year.
The magic word here is "continuously". When this appears, we use the formula :
Where A = The final amount
P = The initial amount
e = The natural base
R = The interest rate ( change to a decimal )
T = The time in years
So in this case:
Substituting into the formula :
 |
Divide both sides by 1000 to isolate the exponential. |
|
Take the natural log of both sides. |
|
What is the base of natural log? |
|
Divide both sides by 0.05 |
|
This is the exact answer. |
Using our calculator , we get:
Answer: It takes about 13.9 years to double the investment.
6. Now you do one:
If $1,000 is deposited today for 10 years compounded continuously, what interest
rate will
it take to double the investment? Round your answer to the nearest
tenth of a percent.
Answer: An interest rate of 6.9% will double the investment in 10 years.
