Domain of Logarithmic Functions
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Domain of Logarithmic Functions

– [Phil] This video we’re
gonna talk a little bit about finding the domain
of a logarithmic function and how to identify
its vertical asymptote. So, in general, if we
have Y equals log of X or natural log of X, or log of any base of X, the domain is that X has
to be greater than zero. And the graph looks something like this. So, it’s only got a
domain of positive values and it’s got a vertical asymptote at X equals zero. So, the Y axis is a vertical asymptote. Now, what’s gonna happen is we’re gonna look at
logarithmic functions that are shifted left and right and that changes where
the vertical asymptote is. For instance, we had the
function Y equals log of X plus two, that would shift our graph
two units to the left. Then that shifts the vertical asymptote two units to the left and so on. Now, what we need, remember, we cannot take log of zero or negative numbers, so to find the domain what we could do is set X plus two, we know that that has
to be greater than zero and solve that for X. So, we get X has to be
greater than negative two. Now, that place that makes it zero, in this case, negative
two is what makes it zero is our vertical asymptote, so the vertical asymptote in this case is X equals negative two. Now, this process will work as long as what we have in here is linear and we’ll look at a quadratic
one here in a second. Let’s do another one. So, Y equals log of three minus four X. And we wanna find the domain and we’re just gonna
do this algebraically. We can verify this with
a graphing calculator. We know that we need three minus four X to be greater than zero. So, let’s see if we can get our domain. That means we need three, we’re gonna add four X to both sides to be greater than four X and now divide both sides by four and we get that X has to be less than 3/4 or if we wanted it in interval notation, it would be negative infinity to 3/4 not including 3/4. Now again, our vertical
asymptote is at the point that makes a zero in the log, so in this case it’s at X equals, happens at 3/4. So that’s how we can look at the domain and identify a vertical asymptote wanna do a little bit
more interesting example where maybe we have a
quadratic in our logarithm and we gotta determine the domain. Let’s say we have and again,
these are logs of any base, I just happen to be doing base 10. Let’s say we have log of X squared minus four and we wanna find the domain. Well, we know we need X squared minus four to be greater than zero. Now, in this case, I
don’t recommend solving it as if it were linear like we did in the last ones. We need to figure out
where X squared minus four is greater than zero. Well, first thing we’re gonna do is figure out where it is zero. So, what we’ll do is
make a little number line for X squared minus four. And let’s see, where is X
squared minus four equal to zero? Well, this guy will
factor into X minus two times X plus two equals zero, so we have zeros
at two and negative two. So, I’m gonna put those on my line and now remember, we do
not want to include zero in this case because log
of zero is undefined. Now, the reason I did this is because I’m gonna pick where my function is positive
and where it’s negative. All I need are test points. So, for instance, if I were to use zero any
place in between here, it’s always gonna have the sign same because we pick the
points which switch signs. If I put zero in here, zero minus four is negative, so it’s negative in between
negative two and two. Let’s pick a point to the left. Say negative three. If I put negative three in and square it I get positive nine minus four is positive, I don’t care
what it comes out to be, just care if it’s positive or negative. If I get positive three I
also get a positive number and so, now from here I can get my domain. My domain is gonna be negative
infinity to negative two because it’s positive and I don’t wanna include negative two because that’s where it’s zero. I have another place where it’s positive, so I’m gonna union those together with a U and that happens from two, not including two to infinity. So, there is my domain of
my logarithmic function. Now, for vertical asymptotes, we actually have two ’cause there’s two places where we can get a zero in the log. They’re gonna occur at negative two and we’re also gonna
have one at positive two, so it’s very possible to have more than one vertical asymptote if you have more than one value that makes it zero inside
the argument of your log.


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