Now that we've looked at exponential functions and how they
behave,
let's look at their inverse, called logarithms. Let's consider this
exponential equation y equals B to the X. Now, let's flip the sides. Still the same
function, isn't it? And I'm going to change the representative letters to
make things a little easier for you.
Okay, I'm going to make it b to the e equals n, okay, rather than b to the x
equals y and I'll tell you why. I'm going to say B is the base which we had before.
e is the exponent that seems logical and, and is just a number.
Ok, let's speak in plain English. How about that? Now
logarithms sounds like a fancy word, doesn't it?
Logarithms are just exponents; means the same thing. We could say here the
exponent, using base b for number n is e, couldn't we?
Let's look at it again. The exponent using base b for the number n is e.
Well,
we can also write it this way,
read the logarithm using base b for the number n is e, is the exponent e
because
exponents are logarithms. They're the same thing. Exponents are logarithms.
You've got to get that down. So the logarithm equals the exponent. This is
two forms of writing logarithms. One is, I'll call, exponential form on the top.
The other one, I'll call, logarithm form. Now there's going to be, in every
problems can be, one easier way to write them. You need to be able to write it
both ways and going from one form to the other I'll call the old switcheroo.
Remember again, exponents are logarithms and of course logarithms are exponents.
Now since logarithms are exponents, they're going to follow the same rules
as exponents. Guess why we just reviewed the rules of exponents? Because they're
the same rules of logarithms.
Now, let's practice this old switcheroo.
Let's rewrite 2 to the 4th equals 16 in logarithm form. Right now it's an
exponential form. We would write the logarithm using base 2 to get the number
sixteen is four. Let's try a backward switcheroo. We have the logarithm using
base 5 to get the number Y is X. Well, we know the base is 5 and whatever the
exponent is, the logarithm is the X so we would rewrite this y equals 5 to the X
the number y is the base 5 to the exponent X.
Let's go back and forth. The number 5 to the exponent 3, sorry, the base 5 to the
exponent 3 equals the number 125 could be rewritten: the logarithm using base 5
of the number 125 is the exponent course a logarithm is an exponent of 3
Now why would we want to know how to do the old switcheroo?
Well, it isn't dumb when you have to solve equations. If we try to solve this
equation in its current form we might have a little trouble.
Let's pull the old switcheroo and see if it doesn't look easier. Let's rewrite
this. That is now in logarithmic format.
Let's write it in exponential form. We go right to the base. We know we're
going to write the base first
so eight
to the x or 8 to the what power is 64?
Well, we know the answer is two.
What about this one? Let's rewrite it in exponential form. The base four to what
power...recall that the logarithm is the exponent...the base four to what exponent
is 64? Make this look good.
This isn't so bad, is it?
Let's switch things around a little bit. We'll still pull the old switcheroo.
Hopefully it'll be easier. Now we go to the base 5 to the what power. Remember
that the log is the exponent so this time we're going to go five to the third
power equals the number X. Well what is 5 to the 3rd power?
Not too bad. Now if you just restate it using the old switcheroo. Now since
these are exponents we're going to have to remember all the rules of exponents
so
remember the rules of negative exponents. x to the negative a equals not just the
opposite but rather the multiplicative opposite, the reciprocal. So X to the
negative a equals 1 over X to the a. Remember that.
So, using that, let's try and solve this logarithmic equation by rewriting it in
exponential form. The base is three; the exponent is X ;the number is one-ninth. What
do I have to raise 32 to get one-ninth or 1 over 3 squared because 3 squared is
9? I want one over three squared; we want to raise it to the negative twoth
because that'll give us the reciprocal. Ok, you have to know all the rules of
exponents here to solve logarithmic equations.
Let's try another one. The logarithm base 4 of on-fourth equals x. Well, if we rewrite it
four to the x or four to the what equals one-fourth we've got negative 1 is the
answer because four to the minus one is the reciprocal of four or one-fourth now.
Practice makes perfect.
Let's do another. We're going to rewrite it in exponential form.
10 to the what equals 1 over a hundred? Well, it's not so bad, is it?
Remember ten to the twoth is a hundred and we want one over a hundred so the
answer was negative 2. If you use negative in an exponent you get the
reciprocal. Now there's some other things about exponents that we're going to need
to remember. The "a"th root of x written as an exponent is X to the 1 over a.
That's going to be important. For instance, X to the one-half is
the square root of x. Remember? We don't write the two but you have to know that.
Guess what X to the one-third is? It's the cube root of x.
Okay.
x to the one-fifth would be the fifth root of x so this is all rules of exponents
are going to come into play when we do and work with logarithms.
So, with that said, the logarithm using base 25 of 5. Well, let's rewrite it.
What do I have to raise 25 to, to get to 5, to get to a smaller number. I'll give you
a hint.
You've got to take the square root so the exponent here is going to be a
representative of the square root, one-half. I want you to relax your mind
here a little bit because if you put this calculation log base 25 into your
calculator quite likely you're not going to get a half. Quite likely you're going
to get a decimal approximation .5. So logarithms while they have a very logical
meaning here and they're related to exponents. They're going to be very
practical in real life and probably you're going to want to look at them as
uh, decimal approximations.
Now, let's rewrite this one log base X. What is the base for the number 243
using an exponent of five? So, what to the fifth power is 243? I'll give you
a hint. It comes out even. What number raised to the fifth power is 243? Well, if you
take the fifth root of both sides (remember how to take the fifth root?) 243 to the
one fifth power .Make sure you put the one fifth in parentheses and you'll get
comes out even I told you. This one isn't so bad. Keep your discipline. Don't get
afraid. Rewrite it.
What's the base? Eight. What's the exponent? X. So 8 to what power is a half?
Wow, 8 to what power is a half?
Well, let's get from 8 to 2 first.
8 to the one-third is two. The cube root of eight is two. We don't want two, do we?
We want one over two so our answer is a reciprocal of that. So our final answer
is negative one-third. Got it? Could even be negative .3333 because that's what
negative one-third is, isn't it?
Okay. We've got to relax our minds here. Let's look at another rule. The...hopefully
you'll recall about exponents.
Remember zero as an exponent? 5 to the second times 5 to the zeroth. We know adding
exponents is 5 to the second. 2 plus 0 is 2 which means what is 5 to the zero have to be if we
multiply 5 to the second by it and get 5 to the second. This expression has got to
equal one, doesn't it? Well, in fact, any number to the zero power is going
to result in one.
Well,
what will the logarithm of 1 always be? Then 7 to what power equal not asking you
what you multiply 7 by I'm saying what do you have to raise 7 to to get one. We
just told you; that's the answer. Think about it 7 to the 0. Anything to the 0 power
equals 1 so the logarithm of 1 will always be 0
regardless of which base you use. That's why I hope you appreciate the graph of
any logarithm function
no matter what base is going to look something like this with the point (1,0) on
it. The point (1,0) because the logarithm of, of any base of one is going to be 0 and
in fact this logarithm graph is always going to be the inverse function of as
we said before the exponential. Remember what that looks like and remember what
it is to be an inverse. An inverse is the mirror about the line y equals x so,
son-of-a-gun, the logarithm is the inverse of the exponential function.
Okay, well anyway,
It's your turn now. Go ahead and practice. Go do your homework.