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How To Solve Exponential Functions & Inequalities Step-by-Step with Real-Life Word Problems
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May 9, 2025
In this video, I break down how to solve exponential equations and inequalities—even if you're just starting out! Whether you're dealing with equations like 2^x = 8 or modeling exponential growth in real-world scenarios (like bacteria populations or compound interest), this lesson has it all. 🔍 What you'll learn: How to rewrite exponential expressions with the same base Solving exponential equations by equating exponents Writing and solving real-world exponential models Using the compound interest formula for financial growth Solving exponential inequalities and understanding key rules Practice problems and examples throughout!
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0:01
we are talking about solving exponential
0:03
equations and then inequalities as well
0:06
But we're going to start with just uh
0:10
equations Okay So we talked about
0:13
graphing
0:14
uh exponential functions Now we're going
0:16
to talk about
0:17
solving
0:19
exponential functions So how do you
0:21
solve
0:22
it
0:23
Now I gave you an example on the board I
0:27
have here 2 to the x is equal to 8 to
0:31
the 3 right And how would you solve this
0:34
Now to solve an exponential equation you
0:37
have to make sure that the base Hey what
0:41
are you doing
0:42
Uh put that away
0:47
It doesn't matter Honesty doesn't mean
0:49
that because she's honest that means
0:52
it's right So 2 to the x is equal to 8
0:55
to the 3 right I'm trying to solve
1:00
this So to do that I have to make sure
1:05
that the base is the same Right We're
1:06
going to apply this an equation that we
1:08
have If I have a b to the
1:11
x is equal to b to the y That means that
1:15
x is equal to y Right So we're going to
1:18
try to transform this equation to get to
1:21
this We want to make sure that the bases
1:23
are the same Right So I have 2 to the x=
1:26
8 the 3r So what am I going to do here
1:28
I'm going to break down
1:31
8 So that way I'm going to turn this
1:33
into two to the
1:36
something in order to be able to solve
1:38
this equation So eight by itself if you
1:43
do your three factor right 8 is what 2 ^
1:48
3 So now we can replace this So now we
1:51
have
1:55
23x right is equal to 23 the whole thing
1:59
to the third power We've done this right
2:01
we've done this kind of stuff So we have
2:03
that Now what I have to do is I can
2:06
solve this equation So this is 2 to the
2:08
x is 2 to the 9th right I can raise this
2:13
to 3 and 3 is 9 So now I can solve this
2:17
equation If 2x is equal to 2 that mean x
2:20
is equal to what 9 It's easy Now I can
2:23
solve this So x is equal to 9
2:26
And then I solve my problem right But
2:29
you cannot do this unless the base of
2:32
the exponential functions are the same
2:34
Equations are the same If they're not
2:36
the same you can't do that So you have
2:37
to make sure you transform it Okay
2:40
Another example I have 9 2x - 1 is equal
2:43
to 6x By just glancing at it what would
2:46
you want to do to solve this equation
2:49
here Um break down nine into three squ
2:53
All right break down 9 and 3 squ right 9
2:55
is 3 and 3 which is 3^ 2 So now we can
2:58
turn this into 3^ 2 to the^ of 2x - 1 =
3:04
3 6x Right And now we can solve this So
3:07
now it's going to give me 3 I'm going to
3:10
multiply I'm going to now have 2 * 2x
3:14
-1 = 3 6x Since the bases are the same
3:19
now I can solve the equation by just
3:22
ignoring the base right So now I'm going
3:25
to have 2 to 2x - 1 is = 6x And then I
3:32
can solve this equation So that is 4x -
3:35
2 is
3:37
6x right And then 4x
3:41
here So we got -2 = 2x and I divided by
3:46
2 we get x = to -1 So for this equation
3:51
x has to equal1 for this statement to be
3:55
true Right So your goal is to make sure
3:59
the base is always the same If as long
4:02
as the base is the same you can solve
4:04
the equation If the base is not the same
4:07
you have to make it the same in order to
4:09
solve the equation Now let me give you
4:10
another one that you can try on your own
4:12
So what if I have 5 to 5x right is
4:16
equal
4:18
to 125 x + 2 Go ahead and take like a
4:22
couple of minutes to solve this 5x is
4:25
equal to 125 to the^ x + 2 How would you
4:29
solve that problem
4:32
5x is 125 to the power of x + 2 Okay you
4:37
want to make sure the basis is the
4:39
same Get the same base before you solve
4:43
it All right
5:39
So what do we have
5:45
What
5:49
All right So you you change this into
5:52
what 5 to the^ of what Three right So 5
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to
5:56
5x is 5 to the^ 3 The whole thing times
5:59
x + 2 like that Correct And then you
6:02
just have 5x is equal to 3x + 6 And then
6:07
solve for x
6:10
right So you got 2x is 6 and then x is
6:13
what Three So x is three here 5^ 15 =
6:18
1.5 ^ 5 It's the same thing right So
6:23
here x is equal to 3 All right Now that
6:26
we know how to solve exponential
6:27
equations and we're going to learn how
6:29
to solve a word problem using
6:31
exponential functions Okay This you need
6:34
to pay attention here because this this
6:36
is an introduction to a new concept
6:38
Right So the first thing that we need to
6:41
remember and we need to know is that
6:43
every exponential function is going to
6:45
be in the form
6:47
y= a to the b x like this This is what
6:52
you're going to be using This is your
6:53
model that you're going to use All
6:55
exponential function are going to look
6:56
like this right Y is equal to a b to the
7:00
x Now given this how do you solve this
7:04
problem So we we we are told that K
7:08
starts an experiment with se 7,500
7:11
bacteria cells right And then after 4
7:15
hours there are 23,000
7:18
cells right They start the experiment
7:21
with 7,500 cells and then after 4 hours
7:24
now you have 23,000 cells So as you can
7:27
see those bacteras are growing
7:29
exponentially They're growing really
7:30
fast This is not a linear growth This is
7:34
an exponential growth Okay Now the
7:36
question is they want you to write an
7:38
exponential function that could be used
7:40
to model the number of bacteria after x
7:45
hours if the bacteria changes at the
7:48
same
7:48
rate Right The bacteria is changing at
7:51
the same rate So the rate at which the
7:53
bacteria is changing is the same So the
7:55
way we're going to solve this problem is
7:56
we're going to we're going to begin with
7:58
the equation y = a bx Right This is the
8:03
expiration model that we're going to use
8:05
and the rate is B Okay the rate is B A
8:10
is your original amount So A is going to
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be the original amount that you have
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original
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amount B is going to be called the rate
8:22
of the rate at which
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uh the spe the bacteria the cells are
8:30
changing are growing the cells are
8:32
growing Right So this is how we're going
8:35
to use now How we going to use this
8:36
information All right So here's what we
8:39
know When the experiment started how
8:42
many bacteria did you have
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70,500 right So original means at time
8:50
zero you have what
8:54
7500 bacteria right At time zero meaning
8:58
when you start experiment this is how
8:59
much you had And after 4 hours what do
9:02
you have After 4 hours you have
9:06
23,000 Right So how do we use this to
9:09
find the equation I'm glad you asked the
9:12
question right So the first thing we
9:14
want to do is we want to find A and we
9:16
also want to find what B Okay So the
9:19
first information is going to help us
9:21
find
9:22
A Okay Because here's the thing when uh
9:26
they're telling us that when when x is
9:29
zero y is equal to what 7500 So I'm just
9:34
going to replace this right 7500 is
9:37
equal to a * b to the power of what 0
9:42
Right I'm going to solve it So b to the^
9:45
of 0 is what Any number that raised to
9:46
the power 0 is what One So therefore
9:50
7500 is equal to what A So now we have a
9:55
So knowing a we can solve for b I know
9:58
for a fact that now I know that y is
10:00
equal to
10:02
7500 time b to the x Now the only thing
10:04
that I need to solve for now is b
10:07
Correct So how am I going to solve for b
10:09
I have another piece of information I'm
10:11
told after 4 hours I have 23,000 sales
10:15
So I'm going to use that to solve for b
10:17
I'm going to replace again here
10:21
23,000 =
10:23
7500 * B to the power of what Four I'm
10:27
going to solve for B here right So how
10:31
am I going to do this now I'm just going
10:33
to put 700 B to the power 4 Because
10:36
after 4 hours I have 7 uh 23,000 cells
10:40
So that can help me solve for B So the
10:43
first thing I'm going to do is divide by
10:45
what
10:46
7500 right
10:49
You going to divide it by 7500 Can
10:50
somebody be kind enough to divide 23,000
10:53
by 7500 See what you get
11:01
23,000 divided by 7500 What'd you get
11:04
Jack
11:08
No No way 23,000 divided by 7500 can't
11:10
be 06 I got 0 I got 3.066
11:16
3.066
11:18
like it's like 6 or it's like 61 3.066
11:22
All right 3.06 like this All right Let's
11:23
just put 6266 Right Let's put 67 right
11:26
Equals to B to the 4th Now how about
11:28
yourself for
11:30
B Y'all remember what we did here
11:34
To the fourth root right So I'm going to
11:36
raise it to the power of what 14 14 And
11:39
that going to give us B right So what's
11:41
that equal
11:50
3.067 to the^ 14 Now B would tell us
11:54
would give us
11:55
3.0 67 Okay I got it 14 What is it It
11:59
says it's 1.3 23 36 3 4 2 3 We can write
12:04
it off to two decimal places Okay 1.32
12:09
32 Right All right So B is equal to 1.32
12:13
So therefore the equation is going to
12:14
look like this Y =
12:17
7500
12:19
1.32 to the^ x So with this equation now
12:23
we can find out how many bacterias we're
12:26
going to have in 5 hours in 10 hours in
12:29
20 hours or in any given time Okay So
12:32
this is how you use this model So you
12:35
can build an equation based on the data
12:37
right We've given the original and then
12:39
we given at a certain time the number of
12:42
bacteria that we have So we can use that
12:44
to build our equation uh completely Okay
12:48
Now another type of problem is also
12:50
compound interest which is actually
12:52
something that you guys going to be
12:54
interested in because you I know
12:55
everybody likes money and then if you
12:57
invest your money you want to know how
12:59
much your money is going to grow Right
13:00
So the compound interest formula is
13:02
interesting I'm going to give it to you
13:04
in general Right So this is called a
13:06
compound interest formula Compound
13:12
right
13:14
This is called a
13:16
compound interest formula What is
13:19
compound
13:20
interest Those that take finance what is
13:23
compound interest
13:26
What is it
13:31
Right
13:33
What is compound interest
13:39
Interest that is what I don't know
13:41
Compound
13:43
What
13:51
Anyway the compound interest is interest
13:54
upon interest Interest that build on top
13:56
of interest right You want that Okay So
14:00
the problem here they telling us like an
14:01
investment account You invest your money
14:04
somewhere right And the account pays you
14:06
4.2% annual interest compounded monthly
14:11
Meaning every month you get that much
14:14
money right They do this every month The
14:16
more time they do the compounding the
14:18
more money you make right So every month
14:21
you get that kind of money coming in for
14:22
you right So if you invest
14:25
$2,500 what would be the balance after
14:27
15
14:28
years right So how much money would you
14:31
have if you invest
14:32
$2,500 So they're asking you if you
14:34
invest
14:36
$2,500 all right after 15
14:40
years at a rate of
14:44
$4.2% right
14:48
Compounded
14:50
monthly right How much money is going to
14:53
accumulate in the account if you invest
14:55
that kind of money You put the money
14:57
there you let it sit and then the bank
15:00
or whoever you investing with is giving
15:02
you the terms of 4.2% So to find the
15:05
amount we just have to use the formula
15:06
we need to know what we are doing here
15:08
right The rate is given to us The rate
15:10
is 4.2% So we have to turn this 4.2% is
15:15
0.042 Right So that's the rate The next
15:18
is the compound period So they're
15:20
telling us that you do this monthly Yeah
15:23
for 15 years So 15 that's 12 right Right
15:26
So M is the compound period is 12
15:30
because you do it every month There are
15:32
12 months in the year So M is 12 M is
15:35
the compound period right And the T is
15:38
the time The time is what 15 years So
15:42
now to find how much money you're going
15:44
to have in there you just have to plug
15:46
in the information P is how much you
15:48
invested P is your principle It's called
15:49
principal right It's the amount that you
15:51
invested So P is
15:54
2500 Now you just need to know how much
15:56
that is So to find the amount that
15:58
you've accumulated you're going to plug
15:59
this in your calculator and go like this
16:02
What is that Is that h right next to in
16:05
the parenthesis right here Yeah No Oh
16:09
that one One plus
16:12
This is one Yeah I thought it was H No
16:14
that's one plus A is going to be 2500
16:18
right
16:20
1
16:21
+42 / 12 to the^ of 15 * 12 which is 300
16:28
Right So if you put this in your
16:30
calculator you're going to see how much
16:31
money you're going to accumulate after
16:33
15 years of investing
16:35
$2,500 So this is how the bank works You
16:38
invest your money and they give you a
16:41
percentage You leave your $2,500 there
16:44
if you have a what you call a CD account
16:46
where you don't touch it You let it sit
16:48
there and the money accumulate You earn
16:50
money on top of money So 20 $2,500 would
16:53
turn into almost according to the book
16:57
$4,688 So you earn about
17:00
$2400 in 15 years of putting $2,500
17:03
somewhere It earns money right Because
17:07
you can keep your money in your house
17:08
It's not going to earn you any interest
17:10
Or you can invest it Yeah What was the
17:12
number
17:15
46.88 Right
17:18
Yeah 46.88 So in in in 15 years your
17:22
$2,500 earns you like 46.88 which is
17:25
about $2,400 I didn't get that Well you
17:28
I don't know how you put it in your
17:29
calculator Are you supposed to put it in
17:31
all at once or section by section When
17:33
you put it into the formula it's
17:35
different But TM solver Yeah that's what
17:38
you get Yeah Don't don't teach them the
17:40
TM solver We keep that for consumer math
17:42
Okay Yeah Let them suffer with that
17:46
do that We don't even use a TF We going
17:48
to use the regular
17:52
T3 Don't let him use the TF
17:56
You're not going to do that All right So
17:59
now we're going to learn how to solve
18:01
inequalities right We did the
18:03
inequalities Now we're going to learn
18:04
how to solve exponential inequalities
18:08
I'm sure you guys are having fun today
18:11
Yeah you are having fun right Do we have
18:14
do we have book today
18:16
Right So now we have
18:19
so uh linear uh exponential inequalities
18:23
honestly are just like uh um equations
18:28
right So if I have
18:30
16 suppose I have 16 to the
18:33
power 16 to the^ of 2x - 3 and I say
18:37
this is less than uh 8 I'm trying to
18:41
solve this right I'm I'm just going to
18:44
solve this just like I solve the
18:46
equation What am I going to do here What
18:48
base do I want Do you want a base of
18:51
what Four won't work A base of what Two
18:55
No Two Two Four is not going to work
18:58
There's nothing that break down 4 eight
19:00
right So base of two So 16 is going to
19:03
be what Two to the power of what Um four
19:07
2 4 and then a is going to be 2 ^ of 3 3
19:11
right so I can change this into 2 4 * 2x
19:15
- 3 is less than what 2 to the^ of three
19:19
and I can just solve the the inequality
19:21
right I drop the bases and I'm just
19:24
going to solve this straightforward so
19:26
it's going to be 4 * 2x - 3 is less than
19:30
3 and I can solve this yeah do you
19:33
change the sign if you divide the
19:35
exponents Right Yeah it applies The
19:38
rules apply All right If you divide by
19:40
negative then you you change the sign So
19:42
I have
19:43
8x - 12 is less than 3 So that means 8 x
19:48
is less than 15 and then x is less than
19:51
15 over 8 And that's your answer right
19:55
So this is how you solve it Now I'm
19:56
going to show you a little wrinkle here
19:58
that we need to like be aware of So what
20:01
if I have
20:03
2 to the x + 2 is greater than or equal
20:07
to 1 32 We haven't seen this example So
20:10
how would you solve this 2 x + 2 is
20:13
greater than or equal to 1 32 What's the
20:16
difference here There's no exponent
20:19
There's no exponent So how would you
20:20
transform this And what kind of base are
20:22
you looking for here
20:28
It's one over right
20:32
All right We're going 30 32 to the^ of
20:34
-1 so that we can solve this be 2x + 2
20:39
right is greater than or equal
20:42
to 32 to
20:44
the1 and then based on this now we can
20:47
solve this right cuz I can change this
20:49
into a power exponent of what
20:52
two right is this 2 to the^ 5 okay so be
20:55
2x + 2 is greater than or equal to 2 5
21:00
*1 one and then I can solve this So be x
21:03
+ 2 is greater than or equal
21:07
to5 and the rest is just a walk in a
21:09
park I'm not doing this You can do it
21:12
All right So this is pretty much how to
21:14
solve equations and inequalities So now
21:18
we're going to jump into the book and
21:20
start working on some problems here All
21:22
right