MATH 155 SUPPLEMENTAL NOTES 4

LOGARITHMS AND EXPONENTIAL OF UNNATURAL BASES

In this set of supplemental notes we will discuss logarithms and exponential functions of bases other than 10 or e. They are called unnatural bases, and sometimes they mimic natural occurrences better than the two bases that are stated above.

THE FUNCTION y = e ^x

FACT: For any numbers a > 0 and x, a ^x = e ^{x ln a}.

From this we can conclude that a function a ^x will behave like the exponential function e ^x. Therefore, the domain for this function will be (-¥, ¥) and the range will be (0, ¥).

THE DERIVATIVE OF y = a ^x

Let us derive the derivative of the function y = a ^x, and to do this we will start with above fact.

FACT: Given y = a ^x, then y' = a ^xln a.

EXAMPLE 1:

SOLUTION:

EXAMPLE 2:

SOLUTION:

EXAMPLE 3:

SOLUTION:

EXAMPLE 4:

SOLUTION:

INTEGRAL OF a ^u

Let us determine what the integral of a ^u will be. Let a ¹ 1, that way ln a ¹ 0. Now, starting from the definition of what the derivative of a ^u is, we will derive the integral.

Dividing both sides by ln a, we get the following.

Integrate both sides with respect to x.

By the Fundamental Theorem of Calculus, we get the following.

FACT:

EXAMPLE 5:

SOLUTION:

EXAMPLE 6:

SOLUTION:

Let u = x ², then du = 2x dx.

EXAMPLE 7:

SOLUTION:

Now, let me digress a bit. This does not fit the above pattern, so what should we do? Skipping the problem is not an option!J Let us do a simple substitution, let u = x ^x. Next question is how do we find du? Well, we will have to do logarithmic differentiation. If you need to review this topic go to supplemental notes2.

u = x ^x ® ln u = ln x ^x ® ln u = x ln x

Now, back to the integral.

EXAMPLE 8:

SOLUTION:

Let u = -x, then du = -dx or -du = dx.

LOGARITHMS WITH BASE a

FACT: For any positive number a ¹ 1, log _ax is the inverse of a ^x.

The easiest way to evaluate log _a x is to convert it into terms of the natural log by using the change of base formula.

FACT:

DERIVATIVE OF log _a x

Let us derive the derivative of the log _a x.

FACT:

EXAMPLE 9:

SOLUTION:

First of all, I will convert the above logarithm into terms of natural logs.

Now I will find the derivative.

Anytime that I have to find the derivative of log _a x, I will convert it into terms of natural logs, then take the derivative of that result. Doing this allows me not to have to remember a formula.

EXAMPLE 10:

SOLUTION:

First of all, I will convert and simplify the function.

Now, I will find the derivative.

EXAMPLE 11:

SOLUTION:

First of all, I will convert this logarithmic function into terms of the natural log, then I will use the properties of logarithms to simplify the expression. If you need to review the properties of logarithms, then check out supplemental notes 1 for the information.

EXAMPLE 12:

SOLUTION:

INTEGRALS INVOLVING log _ax

Any integral involving log _a x should be convert into terms of the natural log, then you will have to probably have to use u-substitution to integrate.

EXAMPLE 13:

SOLUTION:

EXAMPLE 14:

SOLUTION:

Work through these examples of logarithms and exponential functions of unnatural bases. If you have any questions on specific topics, refer back to previous supplemental notes. If you have questions on any of these topics, please feel free to contact me.

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