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Exponential Functions
Topic 2: Solving Exponential Functions
• Determine the solution of an exponential equation in which bases are powers of one another.
• Use technology to determine the solution of an exponential equation in which bases are not powers of one another.
• Graph data, and determine the exponential function that best approximates the data.
• Interpret the graph of an exponential function that models a situation, and explain the reasoning.
• Solve, using technology, a contextual problem that involves data that is best represented by graphs of exponential functions, and explain the reasoning.
Background Information
In order to solve exponential equations, we need to use the exponent laws. Exponent laws are rules that help to simplify or evaluate powers.
Background Information
Example
a) b)
c) d)
e)
3 27 73
2
7
7
3
3
7
7
2
3
7
7
237
3 27 57
3 27
7
3 37
07 1
2 37
171
7
3 27
67
Information
An exponential equation can be solved either algebraically or graphically.
If two exponential expressions with the same base are equal, their exponents are equal. For example, if am=an, then m = n. This fact assists us in solving exponential equations algebraically.
Information
If two exponential expressions cannot be written with the same base, they can be solved graphically. Method 1: Graph the left hand side in Y1 and the
right hand side in Y2. The x-value of their intersection point is the solution to the equation.
Method 2: Move all terms to one side of the equation. Enter the resulting equation into Y1. The x-intercept of the graph is the solution to the equation.
Example 1
Solve each of the following equations. Verify the answers using substitution. a) b)
Solving exponential equations with common bases
3 436 6x x 23(4) 24x
2 3 46 6x x
2 3 46 6x x
4x
2 3 4x x
4x
24 8x
22 32 2x
4 32 2x
4 3x 34x
Both can be written as powers with base 6.
Both can be written
as powers with base 2.
When you raise a power to another power, you must multiply the exponents.
When you raise a power to another power, you must multiply the exponents.
Since the two powers are equal, the exponents must be equal! Now we can solve. Since the two powers are
equal, the exponents must be equal! Now we can solve.
Before we start this one, we can divide both sides by 3 to simplify.
Example 1Solve each of the following equations. Verify the answers using substitution.
c) 3 19 27x
Both can be written as powers with base
3.When you raise a power to another power, you must multiply the exponents.
Since the two powers are equal, the exponents must be equal! Now we can solve.
Before we start this one, we can write the fraction as a constant with a negative exponent.
3 19 27x
3 12 33 3x
2 6 33 3x
2 6 3x
2 9x 92x
Example 1Solve each of the following equations. Verify the answers using substitution.
d) 25 125x
There is a video solution for this question. Click the link!http://youtu.be/Q6oLW7xep1c
Example 2
Solve each of the following equations. Round the answers to the nearest tenth.
a)
Solving exponential equations without common bases
1 12 5x x
You should note right away that there is no common base for 2 and 5. We cannot solve this algebraically!
Graph the left side in Y1 in your calculator.
Graph the right side in Y2 in your calculator.
Continue to adjust your window until the two graphs display clearly. This is a good window setting for this example.The point where these two equation are equal is marked by the intersection of their graphs.
Use 2nd Trace 5: Intersect to find the intersection point.
x = 2.5
Example 2
Solve each of the following equations. Round the answers to the nearest tenth.
b) 5 43 2 5 xx
You should note right away that there is no common base for 3 and 5. We cannot solve this algebraically!
Graph the left side in Y1 in your calculator.
Graph the right side in Y2 in your calculator.
Continue to adjust your window until the two graphs display clearly. This is a good window setting for this example.The point where these two equation are equal is marked by the intersection of their graphs.
Use 2nd Trace 5: Intersect to find the intersection point.
x = 8.9
Example 3
The population of a specific bacteria growing in a Petri dish is modeled by the function , where P(t) represents the number of bacteria and t represents the time in hours after the initial count.a) Determine the value of t when P(t)=324000
algebraically by rewriting the equation with the same base.
Solving an equation in context using more than one method
4324000 12000 3 t
427 3 t
3 43 3 t
3 4t
12t
Divide both sides by 12000.
Write both sides of the equation with a common base (3).
Since the two sides are equal, we know the exponents are equal.
Solve for t by multiplying both sides by 4.
Example 3
b) What does your answer to part a) mean in this context? Explain.
c) Determine the value of t when P(t)=14000 graphically using graphing technology. Round your answer to the nearest hundredth.
In 12 hours, the bacterial count will be 324000.
414000 12000 3 x
Graph in Y1. Graph in Y2.
Window:X: [0, 2, 0.5]Y: [0, 16000, 2000]
Using 2nd Trace 5: Intersect,
x = 0.56 hours
Example 4
Jessica borrowed $7 500 from a bank to buy new equipment for her band. The bank is charging an interest rate of 3.6%/a compounded monthly. Jessica’s monthly loan payment is $400.
Calculating the time to pay off a loan
a) Determine how many months it will take Jessica to pay off the loan. The loan manager gave Jessica the following equation so she could determine how long it would take to pay off her loan 0.9418491271, where n represents the number of months.
Using 2nd Trace 5:Intersect, we can tell it will take 20months for Jessica to pay off the loan.
The two sides cannot be written with the same base so we must solve graphically!!
Example 4
b) How much interest will Jessica pay on the loan?
Jessica is thinking about adding a lighting and sound package to her purchase. This would add $3000 to her loan.
c) How much is Jessica borrowing now?
interest paid total amount paid – amount borrowed
After making 20 payments of $400, she will have paid $8000.
She only borrowed $7500, so she paid $500 in interest [8000 – 7500].
Instead of borrowing $7500, she borrows $10 500.
[7500+3000]
Example 4
d) The loan manager gave Jessica the following equation so she could determine how long it would take to pay off her loan: 0.9195468805, where n represents the number of months. How much longer will it take to pay off this loan instead of the original loan?
Using 2nd Trace 5:Intersect, we can tell it will take 28months for Jessica to pay off the loan.
The two sides cannot be written with the same base so we must solve graphically!!
It will take 8 months longer to pay off this loan.
Example 4
e) How much more interest will Jessica have to pay if she buys the lighting and sound package?
With the original loan, Jessica would have paid $500 in interest. When she adds the lighting and sound package, she pays $700 in interest. This is $200 more interest than if she had stuck with the original equipment loan.
After making 28 payments of $400, she will have paid $11200.
She borrowed $10500, so she paid $700 in interest [11200 – 10500].
Need to Know
An exponential equation can be solved either algebraically or graphically.
If two exponential expressions with the same base are equal, their exponents are equal. For example, if am=an, then m=n.
Solving an exponential equation algebraically: Write both sides of the equation as powers with
the same base, if possible. Set the exponents equal to each other. Solve the resulting equation.
Need to Know
Solving an exponential equation graphically (2 methods): Enter the LHS of the equation into Y1 and the
RHS of the equation into Y2. Find the intersection point of the two graphs. The x-value of this intersection point is the solution.
Move all terms to one side of the equation and graph in Y1. The x-intercept of the resulting graph is the solution.
You’re ready! Try the homework from this
section.