Finding the Log Graph


The way of finding the log graph is common to all logarithmic function. This is because all logarithmic function have essentially the same shape, at least structurally, it only depends on the base of the logarithm.

First, let us recall the logarithmic function with base \(a\), \(\log_a x\). The most typical bases used are for \(a = 10\), in which case we write simply \(\log x\), and the case where \(a = e\), in which case we write \(\ln x\), and we call it the natural log.

For example, the graph of the natural log function, \(\ln x\) is shown below:

Finding the log graph

Now, let us see what happens when we graph together \(\log x\) and \(\ln x\) (this is the log base 10 and the natural log):

Natural log and log base 10

Do you see any similarities? Well, there are some.

Notice that both graphs have the same general concave shape. Also, both graphs cross the y-axis at \(x = 1\) (which is not a surprise since \(\log_a 1 = 0\) for all bases, with \(a > 0\)).

Another thing is that both graphs approach to negative infinity when \(x\) approaches to 0, and to infinity as \(x\) approaches to infinity.

What if we try to graph logarithmic functions with \(0 < a < 1\)?. Check the example below:

Log functions with base less than 1

Do you see any similarities now? Of course.

Notice that both graphs have the same general convex shape. Also, both graphs again cross the y-axis at \(x = 1\), which is expected.

But now both graphs approach to infinity when \(x\) approaches to 0, and to negative infinity as \(x\) approaches to infinity. It is kind of the opposite behavior as when the base \(a\) is greater than 1.


How Do You Make a Log Graph?

Based on what we found in the previous examples, we can throw in some rules you can you use for when you want to make log graphs:

Assume you want to graph the function \(y = \log_a x\), for \(a > 0\). Then:

Step 1: ALWAYS, the logarithmic function crosses the y-axis at \(x = 1\).

Step 2: If \(a > 1\), then the graph is increasing and concave. Also:

\[\lim_{x\to 0^+} \log_a x = -\infty, \,\, \lim_{x\to \infty} \log_a x = \infty\]

Step 3: If \(0 < a < 1\), then the graph is decreasing and convex. Also:

\[\lim_{x\to 0^+} \log_a x = \infty, \,\, \lim_{x\to \infty} \log_a x = -\infty\]

Easy, right??


More About Log Graphs

First of all, knowing how to graph a function is a crucial skill, considering that the graph of a function gives you a LOT of information about it.

In the previous sections we learned how the base of a log affects the graph. The interesting thing is that the shape and behavior of the logarithmic graph depends only on whether \(a > 1\) and \(0 < a < 1\).

Can a log equal a negative number?

Well, we need to specify what we mean by that. First, the base of the logarithmic function cannot be negative. Also, the argument of a logarithmic function cannot be negative either.

BUT, the logarithm of a number can absolutely be negative. For example: \(\ln(1/e) = -1\).

How do you graph inverse log functions?

Well, the first thing you need to know is that the inverse of a log function will always be an exponential function.

So then, the graphing the inverse of a log function is as easy as knowing what is the corresponding exponential and graphing it.

There are other methods too. You can draw the original log graph, and draw a graph that is symmetric to this given log graph, with respect to the 45o straight line \(y = x\).

Or, use the original graph and change the value of \(x\) by the value of \(y\).

This tutorial is oriented towards the graphical properties of the logarithmic function. For the definition and basic log rules, check this one.




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