# How to Graph ln(x)

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• 0:03 The Steps to Solve ln(x)
• 2:51 Logarithmic vs…
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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).

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.

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.

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.

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.

Here's the solution to the problem:

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

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