**Introduction to Logarithm Basics**

There are two types of functions: polynomial and transcendental. A polynomial *of degree k* is a function of the form *p* ( *x* ) = *a* _{0} + *a* _{1} *x* + *a* _{2} *x* ^{2} + ... + *a _{k} x ^{k}*. Such a polynomial has precisely

*k*roots, and there are algorithms that enable us to solve for those roots. For most purposes, polynomials are the most accessible and easy-to-understand functions. But there are other functions that are important in mathematics and physics. These are the transcendental functions. Among this more sophisticated type of functions are sine, cosine, the other trigonometric functions, and also the logarithm and the exponential. The present chapter is devoted to the study of transcendental functions.

**Logarithm Basics**

A convenient, intuitive way to think about the logarithm function is as the inverse to the exponentiation function. Proceeding intuitively, let us consider the function

*f* ( *x* ) = 3 ^{x}.

To operate with this *f*, we choose an *x* and take 3 to the power *x*. For example,

*f* (4) = 3 ^{4} = 3 · 3 · 3 · 3 = 81

The *inverse* of the function *f* is the function *g* which assigns to *x* the power to which you need to raise 3 to obtain *x*. For instance,

We usually call the function *g* the “logarithm to the base 3” and we write *g* ( *x* ) = log _{3} *x*. Logarithms to other bases are defined similarly.

While this approach to logarithms has heuristic appeal, it has many drawbacks: we do not really know what 3 ^{x} means when *x* is not a rational number; we have no way to determine the derivative of *f* or of *g*; we have no way to determine the integral of *f* or of *g*. Because of these difficulties, we are going to use an entirely new method for studying logarithms. It turns out to be equivalent to the intuitive method described above, and leads rapidly to the calculus results that we need.

**A New Approach to Logarithms**

When you studied logarithms in the past you learned the formula

log( *x* · *y* ) = log *x* + log *y* ;

this says that logs convert multiplication to addition. It turns out that this property alone uniquely determines the logarithm function.

Let *l* ( *x* ) be a differentiable function with domain the positive real numbers and whose derivative function *l* ’( *x* ) is continuous. Assume that *l* satisfies the multiplicative law

for all positive *x* and *y* . Then it must be that *l* (1) = 0 and there is a constant *C* such that

In other words

A function *l* ( *x* ) that satisfies these properties is called a *logarithm function* . The particular logarithm function which satisfies *l′* (1) = 1 is called the *natural* *logarithm* : In other words,

For 0 < *x* < 1 the value of ln *x* is the *negative* of the actual area between the graph and the *x* -axis. This is so because the limits of integration, *x* and 1, occur in reverse order: ln with *x* < 1.

Notice the following simple properties of ln *x* which can be determined from looking at Fig. 6.1:

(i) When *x* > 1, ln *x* > 0 (after all, ln *x* is an *area* ).

(ii) When *x* = 1, ln *x* = 0.

(iii) When 0 < *x* < 1, ln *x* < 0

(iv) If 0 < *x* _{1} < *x* _{2} then ln *x* _{1} < ln *x* _{2}.

We already know that the logarithm satisfies the multiplicative property. By applying this property repeatedly, we obtain that: If *x* > 0 and *n* is any integer then

ln ( *x ^{n}* ) =

*n*· ln

*x*.

A companion result is the division rule: If *a* and *b* are positive numbers then

**Examples**

**Example 1**

Simplify the expression

**Solution 1**

We can write *A* in simpler terms by using the multiplicative and quotient properties:

The last basic property of the logarithm is the reciprocal law: For any *x* > 0 we have

ln(1/ *x* ) = −ln *x*.

**Example 2**

Express ln(1/7) in terms of ln 7. Express ln(9/5) in terms of ln 3 and ln 5.

**Solution 2**

We calculate that

ln(1/7) = −ln 7,

ln(9/5) = ln 9 − ln 5 = ln 3 ^{2} − ln 5 = 2 ln 3 − ln 5.

**You Try It** : Simplify ln( *a* ^{2} *b* ^{−3} / *c* ^{5} ).

Find practice problems and solutions for these concepts at: Transcendental Functions Practice Test.

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