UNIT 5 : EXPONENTIAL & LOGARITHMIC FUNCTIONS
LESSON 2: EXPONENTIAL EQUATIONS
Definition: An exponential equation is an equation where the variable is in the exponent.
Examples of Exponential Equations: 2x = 64; 92x – 1 = 729; 500(1.02)x – 1 = 897.56
Theorem: If ax = ay, then x = y. In words: If an exponential equation has the bases equal, then the exponents must be equal.
This theorem gives us our strategy for solving exponential equations, namely convert each side of the equation to a common base.
Note: Exponential equations may also be solved by taking logarithms of both sides of the equation. This method will be discussed in lesson 6
Example 1: Solve for x.
Example 2: Solve for x. Check # b.
Exponential Growth and Radioactive Decay are applications of exponential equations.
A bacteria culture doubles in size every 10 minutes. It’s growth is measured by the following formula:
· A is the number of bacteria after the given time frame
· A0 is the starting number of bacteria
· 2 is the growth factor
· t is the total time elapsed in the experiment
· d is the doubling period
How many bacteria will there be in the culture after 1 ½ hours if there were 20 bacteria in the original culture?
A = ? A0 = 20 t = 1 ½ h = 90
min d = 10 min.
A = ?
A0 = 20
t = 1 ½ h = 90 min
d = 10 min.
Therefore there will be 10 240 bacteria in the culture after 1 ½ hours.
Note: The half-life of a radioactive substance is the period of time a given amount will decay to half of it’s original amount.
The half-life of radioactive radon is 4 days. I t decays according to the formula below:
· A is the mass remaining after the decay period
· A0 is the original mass of radioactive material
· ½ is the decay factor
· t is the total time elapsed
· h is the half-life of the material
If the amount remaining after 40 days is 6.5 g, calculate the original amount.
A = 6.5 g A0 = ? t = 40 days h = 4 days
A = 6.5 g
A0 = ?
t = 40 days
h = 4 days
Therefore the original mass was 6656 g.
Note: An equivalent formula for radioactive decay is: