– [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.

## 62 Comments

## wtfsaywat

Thank you!

## marius666

When you divided by 4 at 2:09 doesn't that change the direction of the inequality? It's suppose to be 3/4 < x?

## DrPhilClark

No, the domain should be x<3/4. Because of the way I solved, adding 4x to both sides first, I divided by 4, not -4 so I did not need to flip the inequality.

## xxxaliendudexxx2

too fast for me :

## xxxaliendudexxx2

@xxxaliendudexxx2 …but after pausing and rewinding a few times, it actually does help. Thank you! 🙂

## Shon9tilR

I was looking for finding the domain of a quadratic logarithm. Thank you!

## Hackmeister0

Great vid man 😀

## PureBread Floof

Completely forgot Logs. Nearing Finals. Thank you 🙂

## sldkjfasljfwaoptiuwa

Thank you. Your explanation of logarithms with quadratic equations was very helpful.

## Sunflower

THANK YOU!!!!!!!!!

## Holdefer27

Thank you!

## rocco harris

very, very helpful. thanks!!

## Brentavious

I'm cramming like a bitch for my final, and you helped me alot. I really appreciate it

## Otilia DSO

Very helpful, thank you!

## Misaki Usui

What if I have g(x) = -log2 x + 5? I don't understand how to get it?

## Hector Nevarez

At 4:39 you explain how the domain is – infinity and -2 "because it's positive". The negative symbols throw me off and I don't see the correlation with anything positive. Shouldn't the domain of log functions then be defined as the set of all real numbers, not just positive?

## DDGDaltonDDG

What about g(x) = ln(x-x^2) ?

## Leander Uka

you din't show the case when the base is x, not very helpful tutorial

## KidGeniuz

very helpful, thank you!!!!

## 007Telecasters

This really helped since I missed pre calc today! Great video! Thank you!

## RealStigman42

I can't believe this random guy I found on the internet explains this stuff better than my actual pre-cal teacher…

## Robert Calaceto

Wow, thank you.

## FrostyJLive

nigga you white as coke

## kenankim

thanks,,this was helpful..u didnt flip the sign tho..anyways thanks

## rich richmond

3/4>x is the same as x<3/4

## abhijeet 007

i guess u missed on condition dat is

if we take1/ LOG10^(1-x) and to find its domain we first take1-x>0

then 1-x not equal to 1.

## kyle hanson

You shouldve done one with a base. I need help finding the domain of log(base 5)(x+7)

## Kaitie Brush

i cannot thank you enough!!! this was verry helpful thankks

## Michael Russick

What if your base is say "7" or "4" does it change how how you find the Domain?

## qxANGELxp

Fuck you, math.

## Aaron Boyle

Saved my life. #SACtomorrowandnoideahowtodothis

## Arthur Thao

Thanks for the video! Your handwriting is great/neat!

## Ialsopop

I have a math professor with a heavy Asian accent and it is very difficult to follow what she is teaching. This was a big help, thank you.

## BlastFromYesterday

How would you find the domain of a log with a rational expression like log((X+2)/(X^2-1)) ?

Do the normal rules of finding the asymptotes and the domain of rational functions apply?

## na

thank you sir, needed the quadratic example at the end. book didn't explain that situation.

## Arnold Christian

Thanks mate!

## July Myers

thank you SO MUCH for this!

## Joshua French

very good demo, helped soo much

## noah deaton

Just signed in to say thanks and to like! lol

## Kieu Truong

thank you!

## currentlyindisguise

how will we solve it log has different base

## TheReaperAnubis

Why do we exclude zero from the domain of a logarithmic function?

## Blanca Corral

THANK YOU!!! THIS WAS EXACTLY WHAT I NEEDED HELP ON.

## Mimi

do we always set it greater than 0?

## CaptBackwards

This one is a example for a review before the big final, Find the domain of the function: g(x) = 6- 7 log Subscript 6 [start fraction x over 7 End Fraction – 8]Right now i'm trying to rewrite the log function into a exponent function but, maybe that is not how you do it.

## CaptBackwards

I just mı̸̸̸̸̸̸̸̸̸̸̸̸̸̸̸̸̨ade you wipe your screen.

## Anthony Buoscio

Dr. Phil don't need no T.V show to be a good math teacher. Thanks man.

## Ronak Jacob

MAN YOU ARE A LIFE SAVER THNX A LOT

## Anjali Raj

You're awesome! Thanks. This helped 😊

## Douglas Toledo

Thank you very much for the video. It was so simple, even though my teacher didn't go over it

## S Shaffi Syed

Helpful

## BEYOUtifulgrin

Thank you so much for this!

## Christoforos Lapathiotis

194k students have teachers who suck at teaching. Our education is lit af.

## тако вторник

So will x always be greater than zero?

## Julie Wing

what about finding the range of a logarithmis function?

## Venkata Sowmya Pathivada

Sir how to calculate domain of log(x^3-x)?

## Aneesh Vaidya

thank you

## Leila Shaye

Thanks!

## Sharmin Aziz

Great job!!! I’m ready for my exam now! ☺️

## Deenesh creations

Quite helpful Thanks

## Joshua D'lima

Subscribed## Arsh Ansari

Huh😑, everyone can solve these simple questions…