Next we're going to
look at how to be
able to solve some problems.
So let's take a look
at a different question
of exponential equations.
If you have a problem
where the numbers work out
pretty nicely--
when we talk about solving
equations sometimes
we have friendly
numbers like this,
where if you look at 2 to
the x equals 8, hopefully
that's pretty obvious.
And it doesn't
take a lot of work
to know that x should be 3.
2 to the third is 8.
But the problem is,
sometimes numbers aren't
all that friendly in math.
And so you might have something
like 2 to the x equals,
well, let's just change the
eight to a nine, like this.
So 2 to the x equals 9.
Now all of the sudden
it's not three.
2 to the third is 8.
2 to the 4 equals 16.
So it's got to be somewhere
between three and four.
But saying it's between 3
and 4 is not very precise.
And if you look back
to the last example
when we were talking about
money, if you want to really
be pretty accurate to
talk about your money,
like you're in a business field
it's pretty important to be
precise and not
just say, oh, it's
kind of between
$10,000 and $100,000.
You might want to have a
precise value in there.
And so you can think
about how do you
be able to find what
specific value of x
you have, such that 2
to the x equals nine.
That's not necessarily
an easy problem
to work through,
and so for that we
need a new kind of a function.
And so we nee to introduce the
logarithmic function, which
is going to essentially undo
the exponential function,
and take that x value
that's up in the exponents
and bring it down so that we
can solve for x without having
to hopefully get [INAUDIBLE].
Exponential functions
and logarithmic functions
are inverses of each other.
And so I want to just
highlight that one property
of inverse functions is
that when you graph them,
they should be symmetric
across a specific line.
Maybe you've learned
about this somewhere else,
but here's the line y equals x.
And inverse functions should
be symmetric across that line.
So when you graph an
exponential function--
let's take the one we
actually had earlier,
f of x equals 2 to
the x, the graph
will go something like this.
And it would just
go up and continue
increasing very quickly.
And so the logarithmic
function that's supposed
to be the inverse of that--
which, remember, the
base of the exponential
has to be the same as the base
of the logarithmic function.
So this would be the
function log base 2,
because the exponential
had a base of two, of x.
This function is going to be
symmetric across that line
y equals x.
And so that function going
to look kind of like this,
and behave very similarly, but
across that line of symmetry
there.
And so we can use the
log with a certain base,
and notice that where the
x is in the log function
is inside of the parentheses.
So it's pretty common for us
for have something like the log
base 3 of x plus 1.
Or log base 10 of x minus 4.
And notice where the
x's are is usually
inside of the parentheses there.
In that case it's
helpful for us to be
able to try to convert
from having logarithms
to kind of an exponential form.
And so here's the main
way we want to do that.
The way that our logarithmic
function is defined
is to say the log
base b of x-- and you
could actually
replace f of x with y.
I'm just going to
cross this out,
because a lot of times we
use f of x now instead of y.
But when we first learned about
f of x is just replacing the y.
And so if you think about where
the y is and where the b is
and where the x is, you
can convert back and forth
from the logarithmic
equation on the left
to this exponential equation on
the right that says b to the y
equals x.
And so on the left you have
a logarithmic equation.
On the right you have
an exponential equation.
And these two equations
mean exactly the same thing.
So this identity
here helps us be
able to convert from
one form to the other.
If we have something like
2 equals the log base 3
of x plus 1, kind of looks
a little bit like something
we had earlier.
You can convert this
to the exponential form
and be able to solve
for the variable
by taking the base down here,
remember that's where b was.
And it would look
like 3 is supposed
to be the base of the
exponential, raised to the 2
power, equals x plus 1.
And if you forgot where
those letters are,
go back to the last
shot where you see where
the y and the b and the x are.
And make sure that you have
them in the right spot,
because the location of
those is very important.
Or, if you had a problem like 3
raised to the x plus 1 equals,
let's say, 50, there's
an exponential equation.
They're trying to
say, what exponent
do you need in order to make
this to be a true statement?
What value of x do
you need to plug in?
We would have
something like, you
could transform this into
being the log base 3 of 50
equals x plus 1.
And again, make sure to keep
in mind where the x and the b
and the y are in
those equations,
because the placement
is very important.
And so being able
to go back and forth
from the exponential
equation to the log equation
and from the log equation
to the exponential equation
is pretty important.
Go back to our practical
example of money for a second.
Let's say that we have this
equation that says, using t.
So f of t equals 3,000 times
1 plus .02 over 4, to the 4t.
And here's our function that
talks about how much money are
we going to have in our account
after a certain amount of time.
But earlier, we asked the
question, how much money will
we have in four years?
And that's all well and
good, but sometimes you
want to know how long
it will take you to get
a certain amount of money.
Maybe you're saving
for retirement,
and you want to know,
how long will it take
you to get to a certain point?
You might say
something like, well,
when will I have
$5,000 in my account?
$5,000 represents
an amount of money,
not of time, so you
don't plug that in for t.
You plug it in for f of t.
And so then the
question would be,
how do you figure out what t is?
How do you figure out
how long it takes for you
to get to that amount of money?
And so you can divide
both sides 3,000.
And you kind of work your way
to being able to solve for t.
This is 5 over 3
on the left side
when you reduce that fraction.
And remember the decimal on the
inside was 1.005, is to the 4t.
And so you can rewrite
this in that log form.
You could say, well, this
means that you have a log
base 1.005 of 5/3 equals 4t.
And then if you divide
both sides by 4,
you could solve for t.
You don't necessarily
know what all this stuff
on the right side means,
the log base 1.005 over 5/3.
It's kind of like,
is that a year?
Is that 10 years?
Is that 50 years?
Is it 100 years?
I don't really
know, so that's what
we're going to talk
about in just a minute.
But you can at least, now
we've kind of solved for t,
or we can solve for t.
Where before it was
up in the exponent
and you really didn't
know what to do with that.
So logarithms are
really helpful for us
to be able to solve
equations when you have
a variable up in the exponent.
