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AP Calculus AB & BC: Help and Review17 chapters | 160 lessons

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Lesson Transcript

Instructor:
*Laura Pennington*

Laura has taught collegiate mathematics and holds a master's degree in pure mathematics.

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.

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 log*a* (*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 log*a* (*x*), in general.

We have that the derivative of log*a* (*x*) is 1 / (*x*ln(*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*) = log*e* (*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 log*a* (*x*).

We find that the derivative of log(*x*) is 1 / (*x*ln(10)).

Now that we know how to find the derivative of log(*x*), and we know the formula for finding the derivative of log*a* (*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 log*M* (*N*) = log*c* (*N*) / log*c* (*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*ā*f*(*x*) is*a*ā*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 log*a* (*x*), which is *a*, to base *e*.

log*a* (*x*) = log*e* (*x*) / log*e* (*a*) = ln(*x*) / ln(*a*) = (1/ln(*a*))ā
ln(*x*)

This tells us to find the derivative of log*a* (*x*), we need to find the derivative of (1/ln(*a*))ā
ln(*x*). Notice that 1/ln(*a*) is a constant, so by our first fact, the derivative of (1/ln(*a*))ā
ln(*x*) is 1/ln(*a*) times the derivative of ln(*x*). Also, by the second fact, we know that the derivative of ln(*x*) is 1/*x*. Therefore, the derivative of (1/ln(*a*))ā
ln(*x*) is 1/ln(*a*) ā
1/*x*, or 1/(*x*ln(*a*)). Ta-da! We have our formula!

We see how the derivative formula for log*a* (*x*) is derived. Pretty neat, huh?

It's great that we know where this formula comes from, but it is definitely a good idea to put this formula to memory. Not only did it allow us to find the derivative of log(*x*), but we can also use it to find the derivative of log*a* (*x*) for any base *a*. For instance, consider our second fact that the derivative of ln(*x*) is 1/*x*. This actually can be proven using this formula.

We know that the base of ln(*x*) is *e*, so we plug *e* in for *a* in the derivative formula to get that the derivative formula of ln(*x*) is 1 / *x*(ln(*e*)). Now, recall that we said the logarithm log*a* (*x*) is equal to the number we raise *a* to get *x*. Therefore, ln(*e*) is equal to the number we raise *e* to in order to get *e*.

Well, this number is actually 1. Thus, we have 1 / (*x*ln(*e*)) = 1 / (*x*ā
1) = 1 / *x*.

As stated in our facts, we get that the derivative of ln(*x*) is 1/*x*, and we see just how useful this formula can be in the study of logarithms and derivatives.

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AP Calculus AB & BC: Help and Review17 chapters | 160 lessons

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