Logarithmic Functions |
Contents: This page corresponds to § 4.2 (p. 330) of the text.
Suggested problems from text:
p. 337 #1, 3, 6, 12, 13, 17, 19, 21, 27, 29, 37, 41, 47, 59, 65, 73, 78
Definition
Graphs
The Natural Logarithm
If a is any positive number other than 1, then f(x) = ax, the exponential function with base a, is one-to-one, and hence has an inverse. For a review of these concepts, see the section on inverse functions. We can see that f has an inverse by looking at its graph and noting that it passes the horizontal line test. In other words, no horizontal line hits the graph of f in more than one point. Take another look at the applet from the last section that shows the graphs of exponential functions.
Drag the point that is initially at (1,2) to see graphs of other exponential functions.
In all cases except when the base is 1, the graph passes the horizontal line test.
Definition
The logarithmic function with base a, written loga(x), is the inverse of the exponential function ax.
Recall that the inverse of a function just undoes what the functions did, and this idea can be expressed through function composition. The fact that loga(x) is the inverse of ax can be expressed with the following two identities.
logarithmic identity 1
logarithmic identity 2
Example 1.
log2 16 = log2 24 = 4, by identity 1.
log5 (1/25) = log5 5 -2 = -2, by identity 1.
Evaluate the following:
(a) log4 16
(b) log3 (1/27)
(c) log4 2 (Hint: Write 2 as a power of 4.)
We saw in the last section that the graphs of exponential functions are easy to sketch, for they all go through the point (0,1) and are increasing everywhere if the base is larger than 1, and decreasing everywhere if the base is between 0 and 1. Furthermore, as we saw in the section on inverses, a function and its inverse have graphs that are reflections of each other through the line y = x. The applet below shows the graph of an exponential function, the line y = x, and the graph of the logarithm function inverse to the exponential. Since the exponential function is initially 2x, the logarithmic function is initially log2. If you drag the point that is initially at (1,2), then the exponential and logarithmic graphs are updated.
The exponential and logarithmic functions base a.
There are several properties of logarithmic functions that follow easily from the definition and are evident from the graphs in the applet above.
(a) Sketch the graphs of 3x and log3 x and the line y = x in the same coordinate plane.
(b) Sketch the graphs of 1.5x and log1.5 x and the line y = x in the same coordinate plane.
(c) Sketch the graphs of 0.5x and log0.5 x and the line y = x in the same coordinate plane.
Use the applet above to check your answers by dragging the point until the correct exponential function is displayed.
In the last section we described the number e and noted that ex is the most important exponential function. It is so important that it is often called the exponential function. It follows that its inverse, the logarithm with base e, is the most important of the logarithmic functions. The logarithm with base e is called the natural logarithm, and it is denoted ln.
Natural Logarithm of x = ln x = loge x
Many scientific calculators have buttons devoted to the natural logarithm and the logarithm base 10 which is also called the common logarithm.
When we write log x, without a subscript, then we mean the common logarithm, log10 x.
The makers of many calculators follow this convention. For example, the TI-82 has a button LN for the natural logarithm and a button LOG for the common logarithm. The Java Calculator uses the function ln() and log() for natural logarithm and common logarithm respectively.
Use your favorite calculator to verify that ln(e) = 1 and log(10) = 1. These are special cases of the statement loga a = 1.