Unit 1: Relationships between Quantities and Reasoning with Equations
Lesson 7- Writing Exponential Equations
Objectives:
 I can determine if an equation is exponential.
 I can use the properties of real numbers and equality to
solve exponential equations.
 I can determine the constraints placed on the variable in
real-world models.
 I can solve real-world problems by writing an equation that
represents the situation and solving it.
Warm Up: Write an equation that represents the situation and use it
to solve.
1) Barbara works as a babysitter. She charges a $10 fee to cover her
transportation costs plus $13 per hour to watch one child. Write an
equation that gives her total pay, y, for a job lasting x hours.
If Barbara babysat one child for 6 hours, how much money would she
make?
Writing Exponential Equations:
In an exponential equation, the variable is the exponent of a
constant, which is the base.
6s = 216 is an exponential equation because the variable, s, is the
exponent of a constant base, 6.
212 = t4 is not an exponential equation because the variable, t, is the
base of the expression and the exponent, 4, is a constant.
Give an example of an exponential equation and an equation
that contains an exponent but is not exponential.
Exponential:
Not Exponential:
The solution to an exponential equation in one variable is the value
that, when substituted into the equation for the variable, results in a
true statement.
The same properties of numbers and properties of equality used to
isolate the variable in a linear equation can be applied to solve
nonlinear equations as well. For simple exponential equations, you
can sometimes find the solution by usig your knowledge of the
powers of the base. Consider the equation 422x + 1 = 17. To solve,
begin by isolating the term containing the variable, which is the
exponential term.
Don’t forget…
We can rewrite a number in exponential form.
49=
121=
8=
27=
16=
or
81=
or
Let’s try to solve a few…
422x + 1 = 17
232x + 4 = 22
Now you try!
252x - 1 = 49
½ (2x) + 4 = 20
Exponential equations can be used to model situations in which a
rate changes in a uniform way. Suppose that a business currently has
5 employees and the number of employees at the business doubles
every year. The number of years it will take before the company has
80 employees can be found by solving the equation 52x = 80.
Just as in a linear equation, the values of a variable in an exponential
equation can sometimes be limited. For real-world situations, it is
always important to remember what a variable stands for and to set
appropriate limits. These are called constraints.
Some examples of constraints are…
*When counting objects, you cannot have negative numbers.
*Sometimes fractional values are appropriate- such as
measurements of length or weight- and sometimes only whole
number values will work- such as numbers of cats or dogs.
Can you think of any other examples of problems that would
have constraints?
Now it is time to solve some problems.
1) Tessa raised the number 3 to a power and then added 19 to the
result. She obtained the sum 100. To what power did she raise 3?
Did you check the solution?
2) The number of bacteria in a petri dish doubles each hour. At the
start of an experiment, there were 300 bacteria in the dish. When the
scientist checked again, there were 4,800 bacteria. How much time
passed?
Did you check your solution? Does your answer make sense in this
problem?
Examine the two examples that we just solved. Can you
see a pattern or method you can use to help you setup
exponential models? Explain what you see.
Your turn!
Write an exponential equation to model the situation. Then solve
the problem.
Jeremiah raised 4 to a power, multiplies the result by 5, and then
added 1. The result was 321. To what power did Jeremiah raise 4?
The number of fruit flies in a population doubles every day. Josh
collected 6 fruit flies for an experiment. After a certain number of
days, the colony had grown to 48 fruit flies. How much time had
past?
====================================================
Solve.
252x + 1 = 801
Setup and solve.
Jordan raised the number 3 to the power of twice a number n.
She then added 5. The result was 14. What was the value of
n?
Name: _________________________
Unit 1 Lesson 7: Writing Exponential Equations
Solve the exponential equation.
1) 3002h = 4,800
2) ½(4x) – 3 = 29
3) 62x = 48
4) 54x + 1 = 320
Write the exponential model for the situation and then solve.
5) Cole raised the number 2 to the power of twice a number n. He
then added 9. The result was 13. Explain how you can find the value
of n.
6) The number of cells in a sample doubles every minute. A doctor
started with a sample of 25 cells and predicted that, after 5 minutes,
he would have 32 cells. Is his prediction reasonable? Explain.
7) A population of wild hares doubles in size each month. The
number of hares after m months can be describes by the expression
222m. Interpret the meaning of the constant 22.
Use the expression above to find out after how many months would
there be 704 hares?
8) The number of bacteria in a sample quadruples every 24 hours.
There were 20 bacteria in the sample initially. Write an equation to
find b, the number of bacteria in the sample after d days.
Now use the model to find out how long it will take to have 81,920
bacteria in the sample.