In previous sections, we introduced the idea of an inverse function. The fundamental idea is that has an inverse function if and only if there exists another function such that and “undo” one another’s respective processes. In other words, the process of the function is reversible to generate a related function .

More formally, recall that a function (where goes from a set to a set ) has an inverse function if and only if there exists another function going from to such that for every in the domain of and for every in the range of . We know that given a function , we can use the Horizontal Line Test to determine whether or not has an inverse function. Finally, whenever a function has an inverse function, we call its inverse function and know that the two equations and say the same thing from different perspectives.

Let be the “powers of 10” function, which is given by .

a.
Complete the following table to generate certain values of .
b.
Why does have an inverse function?
c.
Since has an inverse function, we know there exists some other function, say , such that writing “” says the exact same thing as writing “”. In words, where produces the result of raising to a given power, the function reverses this process and instead tells us the power to which we need to raise , given a desired result. Complete the table to generate a collection of values of .
d.
What are the domain and range of the function ? What are the domain and range of the function ?

The base- logarithm

The powers-of- function is an exponential function with base . As such, is always increasing, and thus its graph passes the Horizontal Line Test, so has an inverse function. We therefore know there exists some other function, , such that writing is equivalent to writing . For instance, we know that and , so it’s equivalent to say that and . This new function we call the base logarithm, which is formally defined as follows.

Given a positive real number , the base- logarithm of is the power to which we raise to get . We use the notation “” to denote the base- logarithm of .

The base- logarithm is therefore the inverse of the powers of function. Whereas takes an input whose value is an exponent and produces the result of taking to that power, the base- logarithm takes an input number we view as a power of and produces the corresponding exponent such that to that exponent is the input number.

In the notation of logarithms, we can now update our earlier observations with the functions and and see how exponential equations can be written in two equivalent ways. For instance,

each say the same thing from two different perspectives. The first says is to the power , while the second says is the power to which we raise to get . Similarly,

If we rearrange the statements of the facts, we can see yet another important relationship between the powers of and base- logarithm function. Noting that and are equivalent statements, and substituting the former equation into the latter shows, we see that

In words, the equation says that “the power to which we raise to get , is ”. That is, the base- logarithm function undoes the work of the powers of function.

In a similar way, we can observe that by replacing with we have

In words, this says that “when is raised to the power to which we raise in order to get , we get ”.

We summarize the key relationships between the powers-of- function and its inverse, the base- logarithm function, more generally as follows. Let and .

  • The domain of is the set of all real numbers and the range of is the set of all positive real numbers.
  • The domain of is the set of all positive real numbers and the range of is the set of all real numbers.
  • For any real number , . That is, .
  • For any positive real number , . That is, .
  • and .

The base- logarithm function is like the sine or cosine function in this way: for certain special values, it’s easy to know by heart the value of the logarithm function. While for sine and cosine the familiar points come from specially placed points on the unit circle, for the base- logarithm function, the familiar points come from powers of . In addition, like sine and cosine, for all other input values, (a) calculus ultimately determines the value of the base- logarithm function at other values, and (b) we use computational technology in order to compute these values. For most computational devices, the command produces the result of the base- logarithm of .

It’s important to note that the logarithm function produces exact values. For instance, if we want to solve the equation , then it follows that is the exact solution to the equation. Like or , is a number that is an exact value. A computational device can give us a decimal approximation, and we normally want to distinguish between the exact value and the approximate one. For the three different numbers here, , , and .

For each of the following equations, determine the exact value of the unknown variable. If the exact value involves a logarithm, use a computational device to also report an approximate value. For instance, if the exact value is , you can also note that .

a.
b.
c.
d.
e.
f.
g.

The natural logarithm

The base- logarithm is a good starting point for understanding how logarithmic functions work because powers of are easy to mentally compute. We could similarly consider the powers of or powers of function and develop a corresponding logarithm of base or . But rather than have a whole collection of different logarithm functions, in the same way that we now use the function and appropriate scaling to represent any exponential function, we develop a single logarithm function that we can use to represent any other logarithmic function through scaling. In correspondence with the natural exponential function, , we now develop its inverse function, and call this inverse function the natural logarithm.

Given a positive real number , the natural logarithm of is the power to which we raise to get . We use the notation “” to denote the natural logarithm of .

We can think of the natural logarithm, , as the “base- logarithm”. For instance,

and The former equation is true because “the power to which we raise to get is ”; the latter equation is true since “when we raise to the power to which we raise to get , we get ”.
Let and be the natural exponential function and the natural logarithm function, respectively.
a.
What are the domain and range of ?
b.
What are the domain and range of ?
c.
What can you say about for every real number ?
d.
What can you say about for every positive real number ?
e.
Complete the following tables with both exact and approximate of and . Then, plot the corresponding ordered pairs from each table on the axes below and connect the points in an intuitive way. When you plot the ordered pairs on the axes, in both cases view the first line of the table as generating values on the horizontal axis and the second line of the table as producing values on the vertical axis label each ordered pair you plot appropriately.

or logarithms in general

In the previous sections, we looked at two specific (and the most common) types of logarithms, base-10 and natural log. In order to fully discuss logarithms, we need to talk about logarithms in general with any base. Let . Because the function has an inverse function, it makes sense to define its inverse like we did when or . The base- logarithm, denoted is defined to be the power to which we raise to get .

Revisiting

In earlier sections, we saw that that function plays a key role in modeling exponential growth and decay, and that the value of not only determines whether the function models growth () or decay (), but also how fast the growth or decay occurs. Furthermore, once we introduced the natural base , we realized that we could write every exponential function of form as a horizontal scaling of the function by writing

for some value . Our development of the natural logarithm function in the current section enables us to now determine exactly.

In modeling important phenomena using exponential functions, we will frequently encounter equations where the variable is in the exponent, like in the example where we had to solve . It is in this context where logarithms find one of their most powerful applications.