Copyright

How to Graph ln(x)

An error occurred trying to load this video.

Try refreshing the page, or contact customer support.

Coming up next: Defining and Graphing Ellipses in Algebra

You're on a roll. Keep up the good work!

Take Quiz Watch Next Lesson
 Replay
Your next lesson will play in 10 seconds
  • 0:03 The Steps to Solve ln(x)
  • 2:51 Logarithmic vs…
Save Save Save

Want to watch this again later?

Log in or sign up to add this lesson to a Custom Course.

Log in or Sign up

Timeline
Autoplay
Autoplay
Speed Speed

Recommended Lessons and Courses for You

Lesson Transcript
Instructor: Laura Pennington

Laura received her Master's degree in Pure Mathematics from Michigan State University. She has 15 years of experience teaching collegiate mathematics at various institutions.

This lesson will explore the properties of the function ln(x). We will use these properties to graph ln(x), and we will take it a bit further and look at how we can use the relationship between ln(x) and e^x to graph ln(x).

Steps to Solve ln(x)

We are going to use the properties of logarithms to graph f(x) = ln(x). A logarithmic function has the form f(x) = log a (x), and log a (x) represents the number we raise a to in order to get x. We call a the base of the logarithmic function. The function f(x) = ln(x) is a logarithmic function with base e, where e is an irrational number with value e = 2.71828 (rounded to 5 decimal places). Instead of writing the natural logarithm as log e (x), we use the notation ln(x).


lnx1


We are going to use the following properties of the graph of f(x) = log a (x) to graph f(x) = ln(x).

  • The x-intercept, or where the graph crosses the x-axis, of the graph is (1, 0).
  • The y-axis is a vertical asymptote of the graph. In other words, the graph approaches the y-axis, but does not touch it.
  • The domain of the function is all real numbers strictly greater than 0.
  • The range of the function is all real numbers.
  • If the base of the function is greater than 1, then the function is increasing, or rising from left to right, and takes on the following general shape.


General Shape When Base is Greater Than 1
lnx2


If the base of the function is greater than 0 and less than 1, then the function is decreasing, or falling from left to right, and takes on the following general shape.


General Shape When Base is Between 0 and 1
lnx3


Let's examine these properties for f(x) = ln(x). We know the base is e, and e > 1. Therefore, the function is increasing and takes on the general shape shown above when the base is greater than 1. Furthermore, we know that the graph passes through the point (1, 0) and that it approaches the y-axis, but never actually touches it. Lastly, because the domain is all real numbers strictly greater than 0, and the range is all real numbers, we know the entire graph will fall to the right of the y-axis. These facts give us an idea of what the graph will look like.


Rough Sketch Using Properties of Logarithms
lnx4


To be more exact, we can plot a few strategic points, so we know the graph is accurate. To find points, we choose some strategic values of x plug them into y = ln(x) and find the corresponding y-value.

x-value y = ln(x)
e = 2.7 y = ln(e) = 1
e 2 = 7.4 y = ln(e 2 ) = 2
e 3 = 20.1 y = ln(e 3 ) = 3

Thus, we have the points (2.7, 1), (7.4, 2), (20.1, 3). We plot these along with our other point (1, 0), and connect the dots with a smooth curve that takes on the shape described above.


Accurate Sketch of ln(x)
lnx5


Here's the solution to the problem:

The graph of f(x) = ln(x) is shown.


To unlock this lesson you must be a Study.com Member.
Create your account

Register to view this lesson

Are you a student or a teacher?

Unlock Your Education

See for yourself why 30 million people use Study.com

Become a Study.com member and start learning now.
Become a Member  Back
What teachers are saying about Study.com
Try it risk-free for 30 days

Earning College Credit

Did you know… We have over 200 college courses that prepare you to earn credit by exam that is accepted by over 1,500 colleges and universities. You can test out of the first two years of college and save thousands off your degree. Anyone can earn credit-by-exam regardless of age or education level.

To learn more, visit our Earning Credit Page

Transferring credit to the school of your choice

Not sure what college you want to attend yet? Study.com has thousands of articles about every imaginable degree, area of study and career path that can help you find the school that's right for you.

Create an account to start this course today
Try it risk-free for 30 days!
Create an account
Support