


_{}
UNIT 5 : EXPONENTIAL
& LOGARITHMIC FUNCTIONS
LESSON 2:
EXPONENTIAL EQUATIONS
Exponential Equations:
Definition: An exponential equation is an equation where the variable is in the
exponent.
Examples of Exponential Equations:
2^{x} = 64; 9^{2x
– 1} = 729; 500(1.02)^{x –
1} = 897.56
Theorem: If a^{x} = a^{y}, 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.
Solutions:
Example 2: Solve for x. Check
# b.
Solutions:
APPLICATIONS:_{}
Exponential
Growth and Radioactive Decay are applications of exponential equations.
Example 1:
A
bacteria culture doubles in size every 10 minutes. It’s growth is measured by the following formula:
, where
·
A is
the number of bacteria after the given time frame
·
A_{0}
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?
Solution:
A = ? A_{0} = 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 halflife of a radioactive substance is the period of time a given amount will decay to half of it’s original amount.
Example 2:
The
halflife of radioactive radon is 4 days.
I t decays according to the formula below:
,
where
·
A is
the mass remaining after the decay period
·
A_{0}
is the original mass of radioactive material
·
½ is the decay factor
·
t is the
total time elapsed
·
h is
the halflife of the material
If the
amount remaining after 40 days is 6.5 g, calculate the original amount.
Solution:
A = 6.5 g A_{0} = ? t = 40 days h = 4 days
Therefore
the original mass was 6656 g.
Note: An equivalent formula for radioactive decay is: