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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  $2_{x}=32$ 
With the Same Variable on Both Sides  $2_{2x}=5⋅2_{x}$ 
With the Same Base  $4_{3x}=4_{2x+3}$ 
With Unlike Bases  $3_{x+4}=81_{x}$ 
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.
The $x$coordinate of the point of intersection is $1.$ Therefore, the solution to the equation is $x=1.$ This can be verified by substituting $1$ for $x$ in the given equation and checking whether a true statement is obtained. Since a true statement was obtained, $x=1$ is a solution to the equation. Note that if the point of intersection is not a lattice point, the exact solution may not be easy to find using this method.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.
Suppose that Dominika deposits $$500$ and wants to determine how many years it will take her to have $$800$ in this account from interest alone. To do so, she must solve the following exponential equation.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 $8.$
Fortunately, the answer should be rounded to the nearest integer. Therefore, the solution to the equation is $x≈8.$ This solution can be checked by substituting the value into the given equation.$x≈8$
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.
It is seen that $x=1$ is an exact solution. Conversely, the exact value of the second solution cannot be determined from the graph. However, approximated to the nearest integer, this solution is $x≈0.$ These solutions can be verified by substituting into the given equation. First, $x=1$ will be checked.$x=1$
$a_{m}=a_{m}1 $
$a_{1}=a$
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 $b_{x}=b_{y}$ if and only if $x=y.$
This property will be proven in two parts.
If $b>0$ and $b =1,$ then $b_{x}=b_{y}$ 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.
$b_{a(x)}=b_{c(x)}$
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
$a_{m}1 =a_{m}$
$(a_{m})_{n}=a_{m⋅n}$
$3⋅3a =a$
$(a)b=ab$