There’s an interesting question you might ask after learning the power rule for integration, which says that for every positive integer , is an antiderivative of . This can almost be extended to all integers: for every negative integer less than , is an antiderivative of (although 0 has to be excluded from the domain because is undefined if is negative). But the rule cannot apply to , since applying the rule in the case would involve dividing by zero. The question, then, is how we can find an antiderivative in this case.
Well, since is continuous it must be integrable. It can’t be the case that there simply aren’t any antiderivatives. So let’s just give a name to one of the antiderivatives and see what properties we can discover. By the fundamental theorem of calculus, for every positive real number , is an antiderivative of . The domain here is rather than because it has to be an interval for the definite integral over that domain to be defined. The most natural choice for a positive real number is 1, so the particular antiderivative to whom we’ll give the name will be . What name shall we give it? Well, let’s just pick one completely at random… how about “the logarithm function”? I like the sound of that. And for every positive real number , we’ll write the output of the function as (and call it “the logarithm of
x“), so that
We can make a few simple conclusions from this definition about the properties of the logarithm function.
- The logarithm of 1 is 0, because is a definite integral over an empty interval and therefore is equal to 0.
- The derivative of the logarithm function is .
- The logarithm function is strictly increasing, since for every positive real number , , which is the derivative of the logarithm function at , is positive. Consequently, it is also one-to-one and invertible.
- The second derivative of the logarithm function is .
- The logarithm function is concave, since for every positive real number , , which is the second derivative of the logarithm function at , is negative.
Also, if we want to find the value of the logarithm of any positive real number, well, we can’t find the exact value, but we can approximate it using numerical integration techniques. For example we can find that the logarithm of 2 is about 0.69, the logarithm of 3 is about 1.10, the logarithm of 4 is about 1.39, the logarithm of 5 is about 1.61, and the logarithm of 6 is about 1.79. You might notice that this is the sum of the logarithms of 2 and 3. Is there any significance to this? Well, 6 is the product of 2 and 3, so let’s investigate another product, like 4, which is the product of 2 and 2. Well, the logarithm of 4 is 1.39 which is about 0.69 doubled, which is the logarithm of 2. Not convinced? Calculating to another degree of precision, is 1.386, and is 0.693—again, is doubled. It seems like we have a general rule here that for every pair of real numbers and , the logarithm of is the sum of and . If you want, try to find a counterexample. You won’t have any luck! But how can we prove that this rule really will work for every pair of real numbers?
We simply use the definition of the logarithm as a definite integral and do some clever transformations. By this definition, and . So how can we get from the first to the second? Well, we will have to split the integral into a sum of two integrals at some point. At which point will we split the range of integration? The obvious choice is , because then the first term in the resulting sum will be , i.e. . The remaining term will be , and this must be equal to . Often two integrals can be shown to be equivalent by making a substitution. In this case, the substitution would have to change the limits and into 1 and . This is easy, you just need to divide by . So we let , i.e. , which means , and the integrand changes from to , i.e. , and we are done. I’ve rewritten this process below as a sequence of equations, which you may find easier to understand.
For every positive integer , it is clear by applying this rule that we can also say that is . This turns out to also apply for every rational number . Again, this is proven using a substitution. As an integral, is , while is . In order to get to the latter integral from the first using a substitution, the limits need to change from 1 and to 1 and . This is, again, easy: we just need to raise each limit to the power of . So we let , i.e. , which means , and the integrand changes from to , i.e. . Then the proof is completed by taking the out using the constant factor rule for integration.
These two rules are extremely useful because they essentially turn products into sums and powers into products, making calculations easier. In particular, they make differentiation easier, because using the chain rule, we can show that there is a simple relation between a function and its logarithm , namely
which allows us to find the derivative of in terms of the derivative of . If is a product, is a sum, which is easier to differentiate, and if is a power, is a product, which is easier to differentiate. In fact, logarithmic differentiation can be used to prove the product rule from the sum rule, or the power rule from the product rule. Because for every pair of functions and ,
and for every function and every rational number ,
The key thing about the logarithm function, though, is that it shows us how to generalise the operation of taking powers to account for irrational powers. But I’ll explain this in a later post.