Constructing an exponential equation example

Liam opened a savings
account and put $6,250 in it.

Each year, the account
increases by 20%.

How many years will it take
the account to reach $12,960?

Write an equation that
models the situation.

Use t to represent
the number of years

since Liam opened the account.

So I encourage you
to pause this video

and actually try to do
it on your own first.

Try to write this equation
that models the situation

using the variable t in
the way they described.

And then actually
answer the question,

how many years will it take for
the account to reach $12,960?

Well, let's just think about it.

So t is to represent
the number of years

since Liam opened the account.

So let's just say it's
been 0 years since Liam

opened the account.

How much is he going to have?

Well, he's just going
to have $6,250 in it.

That's how much he starts with.

Now, let's say it's
been one year since he

opened the account.

How much will he have?

Well, he's going to
have $6,250 times--

or let's write it this
way-- plus 20% 6,250.

It grows 20% every year.

So this is how much he
started the year with,

and then he gets another
20% of that 6,250.

If we factor out
a 6,250, this is

equal to 6,250 times 1 plus 20%,
or we could write that as 0.2.

Which is equal to
6,250 times 1.2.

Now, how much is he going to
have at the end of two years?

Well, he's going to have the
same amount that he still

had at the end of
one year, times 1.2.

Because it grew by 20% again.

So he's going to have
the amount that he

had at the end of
one year times 1.2,

which is equal to 6,250
times 1.2, times 1.2.

Which is equal to 6,250
times 1.2 squared.

I think you might see
where this is going.

Or I could even write it like
this, order of operations,

you do the exponent first.

So what about after three years?

So after three years, well,
we're just going to compound.

We're going to multiply
by 1.2 once again.

So then he's going to have 6,250
times 1.2 to the third power.

And so after t
years, we're going

to multiply by 1.2
that many times.

So after t years in his account,
he's going to have 6,250 times

1.2 to the t'th power.

1.2 to the t'th power,
or to the t power.

I don't want get
confused with the thing

that you use to take bites with.

Anyway.

So they say, write an equation
that models the situation.

So we want to figure out
how many years will it

take the account
to reach 12,960?

So we essentially
want to say, when

is the account
going to be $12,960?

Or we could write
12,960, when is

that going to be equal to
6,250 times 1.2 to the t power?

So that's the equation
right over there

that models the situation.

And then we need to think
about how we can actually

go about solving this thing.

Well, a natural thing is
to isolate the t variable.

Let's divide both
sides by 6,250.

So we could get-- and if
we flip the two sides,

we could get 1.2 to the t power
is equal to-- well let me write

this, 12,960 divided by 6,250.

And since they're
both divisible by 10,

why don't we divide
them both by 10?

So it's 1,296 divided by 625.

And there's several
ways that you

could solve this
problem at this point.

One way, if you feel
confident that this

is going to have an integer
answer right over here,

you could literally just
try to use your calculator

and multiply 1.2 enough times
to get whatever number this is.

And so we could do it that way.

And as we'll see, there's
a more systematic way

of doing it once you
learn about logarithms,

and I'll do that at the end.

But I'll do that
last just in case

you haven't been exposed
to logarithms yet.

So you could literally
say-- so let me just

exit out of everything.

So you could literally say OK,
let's see, 1,296 divided by 625

is this value.

So let's see how many times
we have to multiply by 1.2.

1.2 times 1.2 gets
us-- well, that

doesn't get us close enough.

So let's try it three times.

So let's take that same number.

Let's just take
1.2, let's raise it.

Let's raise 1.2.

Let's just do it three times.

Times 1.2 times 1.2.

That still doesn't get us there.

What if we were to multiply
by 1.2 one more time?

Well, that actually
gets us there.

And we just did
this by brute force.

1.2 to the fourth power
will give us this value.

So that's one way, kind
of a brute-force way,

of figuring out that
t is equal to 4.

Another way, this just might be
a little bit less intuitive, it

might jump out of
it, gee, this looks

like some type of a power of 5.

We know that 5 to the first
is 5, 5 squared is 25,

5 to the third is 125,
5 to the fourth is 625.

And so you might recognize
this right over here

is 5 to the fourth.

And it's actually
a little bit harder

to recognize that
this right over here

is 6 to the fourth power.

And this right over here is 6/5.

So we could rewrite this as
6/5 to the t is equal to 6

to the fourth over
5 to the fourth.

Which is the same thing as
6/5 to the fourth power.

So here you'd say, well, 6/5 to
the t needs to be equal to 6/5

to the fourth power.

t must be equal to 4.

Now, this is nice
when you can recognize

that this is something raised
to the fourth power, which

isn't easy to do.

Or if you know
this is an integer,

and you can just keep
multiplying 1.2--

if you know it's a low integer.

But the systematic
way of doing it

is to actually use logarithms.

And there's many
videos on Khan Academy

about how to use logarithms.

But if you're more
concerned with, well gee,

if I just want to figure
out 1.2 to what power

is equal to this thing,
what you would do--

and we prove this
in other videos--

is if you say, look, let's
take the thing that we want 1.2

to some power to be.

Let's take the
logarithm of that.

And actually, you could
take a logarithm any base.

Your calculator tends to have
a natural log, which is base e,

and a log base 10.

We could just take
a log base 10.

So let's do that.

So we'll take the logarithm of
what we want to get to, 2.0736,

and divide that by
the thing that we're

trying to take the power
of to get to this number.

So divided by the
logarithm of 1.2.

And once again, we prove this--
actually, I wanted to divide.

So let me insert
division symbol.

So once again, it might
look like a little voodoo

right here.

We prove it in other
videos, but if you

wanted to use a calculator to
calculate things like this,

because sometimes it won't be
a nice integer number of years.

It might be 3 and
1/2 years, or it

might be 7.1234 years,
whatever it might be.

This will give you a
more precise answer.

So what do you want to get to?

You want to get to 2.0736.

What are you raising
to some power?

1.2.

Divide the log of
the thing you're

trying to get to divided
by the log of what the base

that you're trying to raise to
a power, and you click Enter.

And then you get--
so this is literally

another way of saying that
1.2 to the fourth power

is going to be 2.0736.

So once again, if this
looks like voodoo,

you don't know what
logarithms are,

we have videos on
Khan Academy on that.

But there's multiple
ways to tackle it,

especially this problem
where the answer was

a little bit simpler.