Section 4.1: Introduction to Logarithms

In this section, we want to introduce the concept of and notation for a logarithmic function.  We will then discover the basic characteristics of a logarithmic graph and discuss how to create such graphs by hand.

Introduction to Logarithmic Functions

In the previous chapter, we examined exponential functions, such as f(x) = 2^x.  We we graphed them as continuous functions, so that their range included all positive values.  That means that we should be able to solve equations, such as, f(x) = 32 or f(x) = 18, for x.  Using XFunctions, by selecting the function, f, you can see a plot of this function:

By clicking with the mouse, or experimenting with the "x=" box (or by guessing), you can easily find that f(5) = 32, so the only solution to 32 = 2^x is x = 5.  However, it is much more difficult to solve 18 = 2^x.  

With some effort, you might try to narrow down on a solution by taking a series of estimates:

While  x » 4.17 is a pretty good approximation to the solution, as you might expect, we can never write down all the correct digits of this number.  That is because, just like , p, and e, this is an irrational number.  

In the same way mathematicians were led to invent the square root function and the radical sign to allow us to write down the positive solution to 18 = x³ as , we will write the solution to 18 = 2^x as log₂(18).  We read "log₂" as "log base 2".  Just as the cube root function is the inverse of the cube function, log₂ is the name of the function which is the inverse of f(x) = 2^x (i.e., the exponential function with base 2).  Note: f does pass the horizontal line test, and so is invertible.  While we began working with functions with nice algebraic formulas (e.g., x² + 3x), we proceeded to give them one letter names (e.g., f(x) = x² + 3x), and then names with more than one letter (e.g., id, abs, and int).  This notation is even more daunting, since we are using more than one letter (i.e., log) and a subscript (in this case, 2) to indicate the base.  Just as there are many exponential functions (e.g., 2^x, 3^x, (0.25)^x, that is, one for each possible base), there are many different logarithmic functions, or "logarithms" for short, each of whose name is distinguished by a different subscript (e.g., log₂(x), log₃(x), log_0.25(x)).  

Notice the asymmetry in the naming of the two groups of functions.  While we use a formula to describe exponential functions, we simply use a manufactured name to refer to logarithms.  The connection between the two groups of functions becomes clearer, if we introduce corresponding names for the exponential functions, such as exp₂, exp₃, and exp_0.25, so that exp₂(x) = 2^x, exp₃(x) = 3^x, exp_0.25(x) = (0.25)^x.

Since logarithms are defined as inverse functions for exponentials, we know that they will cancel out with exponential functions.  Specifically:

For all numbers b > 0, and not equal to 1, log_b(x) is the inverse of exp_b(x) = b^x, so that they "undo" one another.  As an arrow diagram, we can indicate this as:

    or    

Algebraically, this means that log_b(exp_b(x)) = x and exp_b(log_b(y)) = y.  This looks more natural, if we use the traditional exponential notation, as log_b(b^x) = x and .  In particular, log_b(y) is the unique solution to y = b^x. 

Note: We have to exclude the base b = 1, since exp₁(x) = 1^x = 1 does not pass the horizontal line test.  

As we have seen, this is exactly the type of algebraic property that we need in order to solve equations involving exponentials.  In addition, we know that their graphs are simply reflections of one another.  We will discuss this more in the next section.

Just as, in general, exponential functions can only be computed using a very difficult formula from Calculus, logarithms require an equally difficult formula.  Unfortunately, your calculator may very well not include a specific button for most logarithms!  This means that the only way for us to understand logarithmic functions is through their properties, namely, in terms of their inverse relationship to exponentials.  We will eventually use this relationship, and properties of exponentials to derive many useful algebraic properties of logarithms.  This will allow us to solve exponential equations and compute values for all logarithmic functions.

Practice using logarithmic functions to express the solutions to exponential equations by completing the following Exercises.  

Graphing Logarithmic Functions

Since we know that logarithms are inverse functions to exponentials, we can immediate construct their graphs by hand, and determine a number of their general properties.  For example, since log₂ = exp₂ ^-1 (where exp₂(x) = 2^x), we can simply take a table for values for exp₂:

to obtain a table of values for log₂:

Since we know that the graph of exp₂:

• has a horizontal asymptote of y = 0,
• crosses the vertical axis at y = 1, 
• curves upwards, and
• as we move from left to right, the graph increases forever from the asymptote.

we can infer that the graph of log₂:

• has a vertical asymptote of x = 0,
• crosses the horizontal axis at x = 1, 
• curves to the right, and
• as we move from bottom to top, the graph moves to the right from the asymptote.

Geometrically, by interchanging the roles of the horizontal and vertical axes, the graph of exp₂:

becomes the graph of log₂:

You can see that this graph does go though the three points, (1/2, -1), (1, 0), and (2, 1), from the table, and has a vertical asymptote corresponding the the "value" of -¥ at x = 0.  From this we can be describe the graph more naturally by saying that:

• it increases as we move from left to right, curving downward.

Notice how the shape of the general graph (increasing, curving downwards) is already suggested by our three points.  This mean that, when "connecting-the-dots", we only need to be careful to have the graph approach the vertical asymptote and go off to "infinity" on the right.

Since exponential graphs tend to grow very quickly (remember how they almost look like a right angle, when you "zoom out" far enough), logarithmic functions grow very slowly.  For example, even when x = 1,000,000,  log₂(x) » 20.  This was the reason why they were first invented by John Napier in 1614.

Note: Using logarithms, Napier was able to convert the very large numbers involved in astronomical calculations into smaller, more manageable numbers (which helped Johannes Kepler to demonstrate his laws of planetary motion in 1628).  In fact, using logarithms, Napier was able to create one of the first primitive "calculators" out of a collection of small rods (which later were later called "Napier's bones").  William Oughtred improved on this to create the "slide rule" in 1632, which was used by scientists and engineers for over 300 years, until it was finally replaced in the 1970's by the first hand-held, digital calculators.

It is important to notice that, since we have already observed how all exponential graphs are related by a horizontal scaling and/or flip, it must be true that:

All logarithmic graphs are related by a vertical scaling and/or flip.  This means that any logarithm may be written as a multiple of any other logarithm.

We will explain this phenomenon in more detail in the next section, when we discuss the algebraic properties of logarithms.

In particular, this means that, since we now know how the graph of log₂ looks, we have a pretty good idea how the graph of all logarithmic functions will look: 

• they could be more or less steep, due to a vertical scaling, and 
• they may either increase from y = -¥ at x = 0 (when the base is greater than 1) or decrease from y = ¥ at x = 0, if it has been flipped (i.e., when the base is less than 1).

You can quickly view a sampling of all types of logarithmic graphs, as the base ranges from b = .2 to 1.8 in the Example of the following applet: 

Note: You may have noticed that we had to express log_k(x) in XFunctions as ln(x)/ln(k); we will explain this formula in the next section.  Notice how there is no graph when the base is 1, since log₁ is not defined (i.e., 1^x is not invertible).

You can create different examples of logarithmic graphs, by entering your own base in the following box and clicking the "Set Base" button:

Select the base, b =

This defines the functions "expb" and "logb" in XFunctions, corresponding to the exponential and logarithm, exp_b and log_b, with the base you specify.  It also creates an Example, showing the two graphs on the same axes.  Try several examples and notice how the two graphs are always reflections of one another.

To summarize what we have seen so far, we can graph any logarithmic function, log_b, as follows:

• Reverse the values in the graph of the corresponding exponential function, exp_b:

  x	y = expb(x)
  -¥ -1 0 1	0 1/b 1 b

  to obtain a table of values for log_b:

  x	y = logb(x)
  0 1/b 1 b	-¥ -1 0 1

• Draw the vertical asymptote at x = 0, plot the points, and  "connect-the-dots"; if b > 1, the graph will naturally curve downwards; if b < 1, the graph will naturally curve upwards.
• While the graph will either increase or decrease as we go the right, it will do so very slowly.  Even though the graph seems to get rather flat, it will not have a horizontal asymptote, but will approach ¥ or -¥.

Since we can obtain a table of points and a rough sketch of the graph, we can then use our transformational graphing technique to construct plots of functions with a logarithmic "core" such as, f(x) = -2log_0.4(x +1) + 3.  This decomposes as:

• add 1 to the input,
• take log_0.4 of this,
• multiply the result by 2,
• negate this, and
• add 3.

We begin with a table of values for exp_0.4(x) = (0.4)^x:

which gives a table of values for log_0.4:

Since b < 1, we know that this graph starts at y = ¥ at x = 0, decreases through the points, (0.4, 1), (1, 0), and (2.5, -1), while curving upwards:

From this, we can perform the following analysis:

We can imagine roughly how the graph will look:

• Shifting the graph up and stretching make the graph look a bit steeper, but the general shape remains the same.
• Once we flip the graph, it will start from an asymptote of x = 0 and go up, instead of down.
• Shifting the graph left will move the vertical asymptote over to x = -1.

Drawing in the asymptote, plotting the points and "connecting-the-dots" gives:

Practice graphing logarithmic functions by completing the following Exercises.  

Go to Properties of Logarithms

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Send questions or comments to jwicks@northpark.edu

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