Defining Logarithmic Functions

In order to analyze the magnitude of earthquakes or compare the magnitudes of two different earthquakes, we need to be able to convert between logarithmic and exponential form. For example, suppose the amount of energy released from one earthquake were 500 times greater than the amount of energy released from another. We want to calculate the difference in magnitude. The equation that represents this problem is [latex]{10}^{x}=500[/latex], where x represents the difference in magnitudes on the Richter Scale. How would we solve for x?

We have not yet learned a method for solving exponential equations. None of the algebraic tools discussed so far is sufficient to solve [latex]{10}^{x}=500[/latex]. We know that [latex]{10}^{2}=100[/latex] and [latex]{10}^{3}=1000[/latex], so it is clear that x must be some value between 2 and 3, since [latex]y={10}^{x}[/latex] is increasing. We can examine a graph to better estimate the solution.

Figure 2

Estimating from a graph, however, is imprecise. To find an algebraic solution, we must introduce a new function. Observe that the graph above passes the horizontal line test. The exponential function [latex]y={b}^{x}[/latex] is one-to-one, so its inverse, [latex]x={b}^{y}[/latex] is also a function. As is the case with all inverse functions, we simply interchange x and y and solve for y to find the inverse function. To represent y as a function of x, we use a logarithmic function of the form [latex]y={\mathrm{log}}_{b}\left(x\right)[/latex]. The base b logarithm of a number is the exponent by which we must raise b to get that number.

We read a logarithmic expression as, “The logarithm with base b of x is equal to y,” or, simplified, “log base b of x is y.” We can also say, “b raised to the power of y is x,” because logs are exponents. For example, the base 2 logarithm of 32 is 5, because 5 is the exponent we must apply to 2 to get 32. Since [latex]{2}^{5}=32[/latex], we can write [latex]{\mathrm{log}}_{2}32=5[/latex]. We read this as “log base 2 of 32 is 5.”

We can express the relationship between logarithmic form and its corresponding exponential form as follows:

Note that the base b is always positive.

Because logarithm is a function, it is most correctly written as [latex]{\mathrm{log}}_{b}\left(x\right)[/latex], using parentheses to denote function evaluation, just as we would with [latex]f\left(x\right)[/latex]. However, when the input is a single variable or number, it is common to see the parentheses dropped and the expression written without parentheses, as [latex]{\mathrm{log}}_{b}x[/latex]. Note that many calculators require parentheses around the x.

We can illustrate the notation of logarithms as follows:

Notice that, comparing the logarithm function and the exponential function, the input and the output are switched. This means [latex]y={\mathrm{log}}_{b}\left(x\right)[/latex] and [latex]y={b}^{x}[/latex] are inverse functions.

In our first example we will convert logarithmic equations into exponential equations.

In the following video we present more examples of rewriting logarithmic equations as exponential equations.

 Convert from exponential to logarithmic form

To convert from exponents to logarithms, we follow the same steps in reverse. We identify the base b, exponent x, and output y. Then we write [latex]x={\mathrm{log}}_{b}\left(y\right)[/latex].

In our last video we show more examples of writing logarithmic equations as exponential equations.

Evaluate Logarithms

Graphs of Logarithmic Functions

Before working with graphs, we will take a look at the domain (the set of input values) for which the logarithmic function is defined.

Recall that the exponential function is defined as [latex]y={b}^{x}[/latex] for any real number x and constant [latex]b>0[/latex], [latex]b\ne 1[/latex], where

• The domain of y is [latex]\left(-\infty ,\infty \right)[/latex].
• The range of y is [latex]\left(0,\infty \right)[/latex].

In the last section we learned that the logarithmic function [latex]y={\mathrm{log}}_{b}\left(x\right)[/latex] is the inverse of the exponential function [latex]y={b}^{x}[/latex]. So, as inverse functions:

• The domain of [latex]y={\mathrm{log}}_{b}\left(x\right)[/latex] is the range of [latex]y={b}^{x}[/latex]:[latex]\left(0,\infty \right)[/latex].
• The range of [latex]y={\mathrm{log}}_{b}\left(x\right)[/latex] is the domain of [latex]y={b}^{x}[/latex]: [latex]\left(-\infty ,\infty \right)[/latex].

In our first example we will show how to identify the domain of a logarithmic function.

Here is another example of how to identify the domain of a logarithmic function.

 Graph logarithmic functions

Creating a graphical representation of most functions gives us another layer of insight for predicting future events. How do logarithmic graphs give us insight into situations? Because every logarithmic function is the inverse function of an exponential function, we can think of every output on a logarithmic graph as the input for the corresponding inverse exponential equation. In other words, logarithms give the cause for an effect.

To illustrate, suppose we invest $2500 in an account that offers an annual interest rate of 5%, compounded continuously. We already know that the balance in our account for any year t can be found with the equation [latex]A=2500{e}^{0.05t}[/latex].

But what if we wanted to know the year for any balance? We would need to create a corresponding new function by interchanging the input and the output; thus we would need to create a logarithmic model for this situation. By graphing the model, we can see the output (year) for any input (account balance). For instance, what if we wanted to know how many years it would take for our initial investment to double? Figure 1 shows this point on the logarithmic graph.

Figure 1

Now that we have a feel for the set of values for which a logarithmic function is defined, we move on to graphing logarithmic functions. The family of logarithmic functions includes the parent function [latex]y={\mathrm{log}}_{b}\left(x\right)[/latex] along with all its transformations: shifts, stretches, compressions, and reflections.

We begin with the parent function [latex]y={\mathrm{log}}_{b}\left(x\right)[/latex]. Because every logarithmic function of this form is the inverse of an exponential function with the form [latex]y={b}^{x}[/latex], their graphs will be reflections of each other across the line [latex]y=x[/latex]. To illustrate this, we can observe the relationship between the input and output values of [latex]y={2}^{x}[/latex] and its equivalent [latex]x={\mathrm{log}}_{2}\left(y\right)[/latex] in the table below.

Using the inputs and outputs from the table above, we can build another table to observe the relationship between points on the graphs of the inverse functions [latex]f\left(x\right)={2}^{x}[/latex] and [latex]g\left(x\right)={\mathrm{log}}_{2}\left(x\right)[/latex].

As we’d expect, the x– and y-coordinates are reversed for the inverse functions. The figure below shows the graph of f and g.

 Notice that the graphs of [latex]f\left(x\right)={2}^{x}[/latex] and [latex]g\left(x\right)={\mathrm{log}}_{2}\left(x\right)[/latex] are reflections about the line y = x.

Observe the following from the graph:

• [latex]f\left(x\right)={2}^{x}[/latex] has a y-intercept at [latex]\left(0,1\right)[/latex] and [latex]g\left(x\right)={\mathrm{log}}_{2}\left(x\right)[/latex] has an x-intercept at [latex]\left(1,0\right)[/latex].
• The domain of [latex]f\left(x\right)={2}^{x}[/latex], [latex]\left(-\infty ,\infty \right)[/latex], is the same as the range of [latex]g\left(x\right)={\mathrm{log}}_{2}\left(x\right)[/latex].
• The range of [latex]f\left(x\right)={2}^{x}[/latex], [latex]\left(0,\infty \right)[/latex], is the same as the domain of [latex]g\left(x\right)={\mathrm{log}}_{2}\left(x\right)[/latex].

A General Note: Characteristics of the Graph of the Parent Function, f(x) = log_b(x)

For any real number x and constant b > 0, [latex]b\ne 1[/latex], we can see the following characteristics in the graph of [latex]f\left(x\right)={\mathrm{log}}_{b}\left(x\right)[/latex]:

• one-to-one function
• vertical asymptote: x = 0
• domain: [latex]\left(0,\infty \right)[/latex]
• range: [latex]\left(-\infty ,\infty \right)[/latex]
• x-intercept: [latex]\left(1,0\right)[/latex] and key point [latex]\left(b,1\right)[/latex]
• y-intercept: none
• increasing if [latex]b>1[/latex]
• decreasing if 0 < b < 1

Figure 3 shows how changing the base b in [latex]f\left(x\right)={\mathrm{log}}_{b}\left(x\right)[/latex] can affect the graphs. Observe that the graphs compress vertically as the value of the base increases. (Note: recall that the function [latex]\mathrm{ln}\left(x\right)[/latex] has base [latex]e\approx \text{2}.\text{718.)}[/latex]

Figure 4. The graphs of three logarithmic functions with different bases, all greater than 1.

In our first example we will graph a logarithmic function of the form [latex]f\left(x\right)={\mathrm{log}}_{b}\left(x\right)[/latex].

Example

Graph [latex]f\left(x\right)={\mathrm{log}}_{5}\left(x\right)[/latex]. State the domain, range.

Show Answer

Before graphing, identify the behavior and key points for the graph.

• Since b = 5 is greater than one, we know the function is increasing. The left tail of the graph will approach the vertical line x = 0, and the right tail will increase slowly without bound.
• The x-intercept is [latex]\left(1,0\right)[/latex].
• The key point [latex]\left(5,1\right)[/latex] is on the graph.
• We plot and label the points, and draw a smooth curve through the points.

Figure 5. The domain is [latex]\left(0,\infty \right)[/latex], the range is [latex]\left(-\infty ,\infty \right)[/latex].

How To: Given a logarithmic function with the form [latex]f\left(x\right)={\mathrm{log}}_{b}\left(x\right)[/latex], graph the function.

1. Plot the x-intercept, [latex]\left(1,0\right)[/latex].
2. Plot the key point [latex]\left(b,1\right)[/latex].
3. Draw a smooth curve through the points.
4. State the domain, [latex]\left(0,\infty \right)[/latex], the range, [latex]\left(-\infty ,\infty \right)[/latex].

Summary

The inverse of a function can be defined for one-to-one functions.  If a function is not one-to-one, it can be possible to restrict it’s domain to make it so. The domain of a function will become the range of it’s inverse.  The range of a function will become the domain of it’s inverse.  Inverses can be verified using tabular data as well as algebraically.

The base b logarithm of a number is the exponent by which we must raise b to get that number. Logarithmic functions are the inverse of Exponential functions, and it is often easier to understand them through this lens. We can never take the logarithm of a negative number, therefore [latex]{\mathrm{log}}_{b}\left(x\right)=y[/latex] is defined for [latex]b>0[/latex].

Knowing the squares, cubes, and roots of numbers allows us to evaluate many logarithms mentally because the logarithm is an exponent.  Logarithms most commonly sue base 10, and often use base e. Logarithms can also be evaluated with most kinds of calculator.

To define the domain of a logarithmic function algebraically, set the argument greater than zero and solve. To plot a logarithmic function, it is easiest to find and plot the x-intercept, and the key point[latex]\left(b,1\right)[/latex].