In this lesson, we will learn a formula that we can use to find the derivative of log(x). We’ll also look at where this formula comes from and use it to find another well known derivative.
Steps to Solve
We want to find the derivative of log(x). To do this, we first need to examine the expression log(x). In general, a logarithm has the form loga (x). That is, we call a the base of the logarithm. Also, loga (x) represents the number we raise a to in order to get x.
Now, notice that log(x) doesn’t have a base shown. When this is the case, the implied base is 10. Therefore, log(x) = log10 (x).
Alright, now we can get to the derivative of log(x). This derivative is fairly simple to find, because we have a formula for finding the derivative of loga (x), in general.
We have that the derivative of loga (x) is 1 / (xln(a)). Wait! What the heck is a natural log of a, notated in our formula as ln(a)? No worries, ln(a) is simply another logarithm with implied base e, where e is the irrational number with approximate value 2.71828. That is, ln(x) = loge (x).
Okay, so let’s use this formula to find the derivative of log(x). We have that the base of log(x) is 10, so we plug this into the derivative formula for loga (x).
We find that the derivative of log(x) is 1 / (xln(10)).
Deriving the Formula
Now that we know how to find the derivative of log(x), and we know the formula for finding the derivative of loga (x) in general, let’s take a look at where this formula comes from. Of course, we will need to go over a few facts first, so let’s get started.
First of all, in order to explain where the formula comes from, we need to be familiar with the change of base formula for logarithms. This is a straightforward formula that allows us to express a logarithm using a different base, and it is logM (N) = logc (N) / logc (M).
Alright, there are two more facts that we’re going to need to know to continue:
If a is a constant, then the derivative of a⋅f(x) is a⋅f ‘ (x).
The derivative of ln(x) is 1/x.
Now, let’s use these facts to derive our formula. First of all, we’re going to use the change of base formula to change the base of loga (x), which is a, to base e.
loga (x) = loge (x) / loge (a) = ln(x) / ln(a) = (1/ln(a))⋅ln(x)
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