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,
logax = 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.
Who invented such a thing?
John Napier, a Scottish mathematician is credited with the invention of logarithms. His book, A Description of the Wonderful Law of Logarithms, was published in 1614. Napier devised a method to facilitate calculations by using addition and subtraction rather than multiplication and division. Today, we ususally use logarithms to the base 10, common logs, or logarithms to the base e, or natural logs. In Napier's publication, he describes logs to the base 2.
Some examples of logarithms
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 logarithmThe 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,
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
As you worked through these exercies, did you notice the outputs of logarithms increase linearly as the inputs increase exponentially?
The natural logarithm of a positive number x, is the exponent you get when you write x as a power of e. Recall that
logex = ln x
ln x = k if and only if e k = x .
Logarithmic calculations you cannot do by hand.
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.
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) = logax.
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) = logex = 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,
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.
In the next section we will describe the properties of logarithms.
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