When we say that logarithms can't be negative, what we mean specifically is that we can't
have the argument of the logarithm be negative.
This is a logarithm.
This little number is called the base.
This number is called the argument.
This is the answer.
The way logs are defined, we're not allowed to make the argument negative.
In fact, the argument of a logarithm can only be a positive number not equal to 1.
To understand why, we have to talk about the base of the logarithm first, before we can
talk about the argument.
By definition, when we say log_b(x)=y, that's the same thing as saying b^y=x.
So if you want to find log_b(x), you're asking "what power do you have to raise b to in order
to get x?".
Or you can say, if I raise b to the power of y, I'm going to get x.
In other words, if we go back to our original logarithm, if we raise 2 to the power of 3,
we're going to get 8.
Or 2^3=8.
That's what the logarithm tells us.
So b, the base of the logarithm, is going to turn into a power function.
We're going to raise b to the exponent y, and get a result of x.
So what kinds of numbers can the base actually be?
Well let's talk about positive numbers, negative numbers, and 0 and 1 specifically.
Well if b is 0, then every one of these logs is undefined, because they turn into these
equations, and we can't solve for a or b or c.
There's nothing we can raise b to that's going to give us 2 or 5 or 10.
Because 0 raised to anything is still 0, so that's why we don't allow 0 to be the base
of a logarithm.
In the same way, if b is 1, we end up with these equations, and again we can't solve
for a, b, or c. 1 raised to anything is still 1.
There's no way we can get to 2 or 5 or 10, so we can't allow logs to have a base 1 either.
Similarly, we don't allow negative bases, because if we want to raise that negative
base to a power other than a positive integer, things can start to get tricky.
So for example, if we have the base -2 for our logarithm, and we then want to solve log_(-2)(x)=1/2,
then we're really solving the equation x=(-2)^1/2, or x=sqrt(-2).
As you know, we can't take the square root of a negative number, so this equation breaks
down.
That's why we don't allow negative numbers in the base of the logarithm.
So to summarize, the base of the logarithm can't be a negative number, it can't be 0,
and it can't be 1.
In other words, only positive numbers other than 1 are acceptable for the base of the
logarithm.
So the question is, why does what we allow for the base of the logarithm effect what
we can allow for the argument?
Because we just said we only allow for positive bases, and we know those bases are going into
the equation b^y=x for b.
Which means we're always going to have a positive number for b.
What we're trying to figure out is why x always has to be a positive number also.
Well the answer is that, no matter what you use for the exponent y, if the base b is a
positive number, then you're always going to get a positive number for the argument
x.
To prove to ourselves that this is true, let's look at what happens when y is negative, positive,
and 0.
Well, when you raise a positive number to a negative number, it just moves that number
to the denominator, and you're still going to end up with a positive number.
When you raise a positive number to the power of 0, you get 1, a positive number.
And when you raise a positive number to a positive number, you're going to get a positive
number.
In other words, a positive number, no matter what you raise it to, is going to be a positive
number.
So because we're saying that the base of the logarithm has to be a positive number other
than 1 in order for the log to be defined, that means b is always going to be positive,
which means that x is always going to be positive, too.
So what we can conclude then is that, in order for the logarithm to be defined, the argument
x always has to be positive, so that we don't run into a problem with our bases.
We can't input a negative number, we can't input 0, and we can't input 1, but we can
input any other positive number.
And that's why the argument of a logarithm can only be a positive number.