## The Biology Project > Biomath > Logarithmic Functions > Definition

## Logarithmic Functions

Definition, Examples, and Properties

Logarithms are encountered throughout the biological sciences. Some examples include calculating the pH of a solution or the change in free energy associated with a biochemical reactions. To understand how to solve these equations, we must first consider the definition of a logarithm.

Definition

The formal definition of a logarithm is as follows:

the base a logarithm of a positive number x is the exponent you get when you write x as a power of a where a > 0 and a ≠ 1. That is,

loga x = k    if and only if    a k = x

The key to taking the logarithm of x > 0 is to rewrite x using base a. For example,

log2 32 = 5

can be rewritten as

2 5 = 32

As you work the examples in this section, notice the outputs of logarithms increase linearly as the inputs increase exponentially. Like scientific notation (link to section), logarithms are used to simplify calculations with large numbers.

Examples

Logarithms, just like exponents, can have different bases. In the biological sciences, you are likely to encounter the base 10 logarithm, known as the common logarithm and denoted simply as log; and the base e logarithm, known as the natural log and denoted as ln. Most calculators will easily compute these widely used logarithms.

Base 10 logarithm

The common logarithm of a positive number x, is the exponent you get when you write x as a power of 10. That is,

log x = k    if and only if    10 k = x

Computing the common logarithm of x > 0 by hand can only be done under special circumstances, and we will examine these first. Let’s begin with computing the value of,

log 10.

According to our definition of the common logarithm, we need to rewrite x = 10 using base 10. This is easy to do because 10 = 101. So the exponent, k, we get when rewriting 10 using base 10 is, k = 1. Thus, we conclude,

log 10 = log 101 = 1.

While this example is rather simple, it is good practice to follow this method of solution. Now try the following exercises.

Test yourself with the following exercises

Natural logarithms

The natural logarithm of a positive number x, is the exponent you get when you write x as a power of e. Recall that

loge x = ln x

therefore

ln x = k    if and only if    e k = x

Calculations with logarithms in other bases.

Now, suppose you were asked to compute the value of log 20. What would you do (or try to do) to get an answer? Do you notice anything different about this problem?

As you most likely noticed, there is no integer k, such that 10k = 20. So, in this case, you will need to rely on your calculator for help. Using your calculator you will find,

log 20 ≈ 1.30.

Remember that this is true because,

101.30 ≈ 20.

Again, test yourself

After completing these exercises you will notice that your answers (outputs) are small relative to your large inputs. Remember that logarithms transform exponentially increasing inputs into linearly increasing outputs. This is quite convenient for biologists who work over many orders of magnitude and on many different scales.

Inverses

Since exponential and logarithmic functions are inverses, the domain of logarithms is the range of exponentials (i.e. positive real numbers), and the range of logarithms is the domain of exponentials (i.e. all real numbers). This is true of all logarithms, regardless of base.

Recall that an exponential function with base a is written as f (x) = a x. The inverse of this function is a base a logarithmic function written as,

f −1 (x) = g (x) = loga x.

When there is no explicit subscript a written, the logarithm is assumed to be common (i.e. base 10). There is one special exception to this notation for base e ≈ 2.718, called the natural logarithm,

g (x) = loge x = ln x.

To compute the base a logarithm of x > 0, rewrite x using base a (just as we did for base 10). For example, suppose a = 2 and we want to compute,

log2 8.

To find this value by hand, we convert the number 8 using base 2 as,

log2 8 = log2 23 = 3,

just as we did for base 10.

Using this methodology, test yourself by computing the following by hand.

Properties of Logarithms

Understanding the properties of logarithms will help you solve equations involving exponentials and logarithms. Recall that the formal definition of a logarithm is

loga x = k    if and only if    a k = x

where a, the base of the logarithm, is greater than zero and not equal to 1 and x is a positive number.

Logarithms have the following properties

 1 loga a x = x for any real number x, 2 aloga x = x for any positive number x. For positive real numbers m, n, and any real number r, and a > 0 and a ≠ 1, 3 loga 1 = 0, 4 loga m + loga n = loga (mn), 5 loga m − loga n = loga (m/n) , 6 loga (mr ) = r loga m.

Changing the base of a logarithm

Once you have simplified a logarithmic equation using the definition and properties of logarithms, you will need to calculate the answer. Some calculators will only compute common logarithms and natural logarithms (a = 10 and a = e, respectively). In these cases you will need change the base. The change of base formula is given by, For calculator convenience, we usually select b to be 10 or e.

For example, log73 can be computed as or as in your calculator.

It is also important to realize that to graph the function y = loga x, where ≠ 10 and a ≠  e, in your calculator, you will need to input this function as, or as, *****

In the next section we will begin an exploration of the graphs of logarithmic functions.

The Biology Project > Biomath > Logarithmic Functions > Introduction

The Biology Project
Department of Biochemistry and Molecular Biophysics
The University of Arizona

February 2005
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