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Here are a few recommended readings before getting started with this lesson.
Dominika has a meeting with her guidance counselor on Monday afternoon to discuss her college plans. She is considering where she wants to apply in a few years. Dominika knows that she wants to study in a large city, but not one that is too big. She has two cities in mind. To decide between them, she is paying close attention to their populations, which are given by two exponential functions.
Here, x is the number of years that have passed since the year 2000. Furthermore, f and g are the populations of each city in millions of people after x years. If they keep growing like this, in what year will the populations be the same? Approximate the answer to the nearest century.
An exponential equation is an equation where variable expressions occur as exponents. As with any kind of equation, there are different types of exponential equations.
Example Equation | |
---|---|
With One Variable | 2x=32 |
With the Same Variable on Both Sides | 22x=5⋅2x |
With the Same Base | 43x=42x+3 |
With Unlike Bases | 3x+4=81x |
With a Rational Base | (21)x=8 |
An exponential equation can be solved graphically.
Now, both functions will be graphed on the same coordinate plane.
The number of solutions to the equation is the number of points of intersection of the graphs.
The solutions to the equation are the x-coordinates of any points of intersection of the graphs. Since these graphs intersect at one point, the equation has one solution.
Dominika's first class on Mondays is economics and personal planning. She is told that a certain savings account earns 6% annual interest compounded yearly.
The graphs intersect at one point, so there is only one solution to the equation.
The x-coordinate of the point of intersection appears to be 8. However, looking closely at the graph, it can be seen that the x-coordinate of this point is a bit greater than 8.
x≈8
Consider each side of the equation as a function. Then graph the functions and find their point of intersection.
The graphs intersect at two points. Therefore, the equation has two solutions, which are the x-coordinates of these points of intersection.
x=-1
a-m=am1
a1=a
Multiply
Add fractions
Solution | Substitute | Simplify |
---|---|---|
x=-1 | 2(3)-1=?35(-1)+37 | 32=32 ✓ |
x≈0 | 2(3)0≈?35(0)+37 | 2≈37⇕2≈2.333333…✓
|
Since true statements were obtained, x=-1 and x≈0 are solutions to the equation.
Graph y=40000(1.25)x and y=120000(0.87)x on the same coordinate plane. What is the x-coordinate of the point of intersection? What does it mean in this context?
The graphs intersect at one point. Although it can be seen that the x-coordinate of the point of intersection is a bit greater than 3, its exact value cannot be determined by graphing.
Since x is the number of years that have passed since the year 2020, it can be stated that the attendance to both events will be roughly the same in 2020+3=2023.
Before discussing how to solve exponential equations algebraically, an important property must be learned.
Two powers with the same positive base b, where b=1, are equal if and only if their exponents are equal.
If b>0 and b=1, then bx=by if and only if x=y.
This property will be proven in two parts.
If b>0 and b=1, then bx=by if and only if x=y.
With this property in mind, a method for solving exponential equations algebraically can be explained.
Let b be a positive number other than 1 and a(x) and c(x) be two algebraic expressions in terms of the same variable. If an exponential equation is or can be written in the following form, then it can be solved algebraically by using the Property of Equality for Exponential Equations.
ba(x)=bc(x)
Dominika decides to make good use of her free period after lunch to do some extra credit math problems.
Unfortunately, she is struggling with solving three exponential equations. Help her understand how to solve the equations algebraically to obtain the extra credit she needs!
x=21
a⋅b1=ba
Rewrite 1 as 22
Add fractions
Calculate power
LHS−4x=RHS−4x
LHS/(-3)=RHS/(-3)
Put minus sign in front of fraction
x=-32
a(-b)=-a⋅b
a⋅cb=ca⋅b
Rewrite 1 as 33
Add fractions
Calculate power
Write as a power
am1=a-m
(am)n=am⋅n
3⋅3a=a
(-a)b=-ab
x=1
Subtract term
Identity Property of Multiplication
Calculate power
a⋅b1=ba
ba=b/2a/2
Solve the exponential equations graphically or algebraically. Whenever necessary, round the answers to two decimal places.
Dominika finishes her Mondays with biology class. She learns that bacteria have the ability to multiply at incredible rates. After studying two bacteria populations, she concludes that their growth can be modeled by two exponential functions.
Rewrite the inequality so that each side is an exponential expression with the same base.
LHS/10<RHS/10
Write as a power
(am)n=am⋅n
a⋅am=a1+m
It is shown that the graph of y=10(1.96)x is below the graph of y=14(1.4)x for x<1, even before the first observation was made at x=0. Therefore, the solution to the inequality is, indeed, x<1.
With the topics seen in this lesson, the challenge presented at the beginning can be solved. Dominika has a meeting with her guidance counselor on Monday afternoon to discuss her college plans. Dominika wants to study in a smaller city. She knows that the populations of two cities in the US are modeled by two exponential functions.
Solve the exponential equation 8(1.0048)x=3.5(1.0067)x.
The x-coordinate of the point of intersection cannot be determined precisely on the graph. However, it does show that, to the nearest hundred years, the x-coordinate is 400. This means that the cities will have the same population around the year 2400, many years after Dominika has finished college.
Since Dominika wants to study in a smaller city, she will talk to her counselor about the city with the smaller population during the period of time when she will be in college. Based on the graph, this means she will apply to a college in City B.