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THE EXPONENTIAL AND LOGARITHMIC FUNCTIONS, LOGARITHMIC DIFFERENTIATION

Exponents. The process of defining the meaning of an is a multi-step process.

Step 1. Define the meaning of the symbol an where a is any real number and n is a positive integer as follows:

a1 = a

a2 = a • a

a3 = a • a • a

.....

an = a • a • a .... to n factors

This gives us a meaning for an when the exponent n is a positive integer. The following laws are valid for an defined as above (m and n are positive integers):

Next we want to give a meaning to such symbols as

Our object will be to define these symbols in such a way that laws 1 through 5 will apply in all cases.

Step 2. Assign a meaning to a0. If we want Law 1 to hold so that

we must assign the value one to the symbol a0.

Def. a0 = 1.

Step 3. Assign a meaning to negative integral exponents. If we wish Law 1 to hold so that

am • a - m = am - m = a0 = 1           (m is an integer, a≠0)

we must agree that a - m = 1/am.

Def. a-m = 1/am where m is an integer.

Def. n-th root. If an = p, where n is a positive integer, we call a an n-th root of p, written There may be more than one real n-th root of p. For example since 22 = 4 and (-2)2 = 4, there are two real square roots of 4, namely 2 and -2.

Step 4. Assign a meaning to fractional exponents. If we wish Law 1 to hold so that

we must define the symbol a½ to stand for a square root of a. To avoid ambiguity we can define it to stand for the positive square root. In general, we define the symbol

to stand for the positive q-th root of a.

It follows from these definitions that

and that the above five laws are valid if m and n are either fractions or integers (i.e. if they are rational numbers), provided a and b are positive numbers. If the exponent is irrational, the power is defined to be the quantity approximated by using rational exponents which approximate the irrational exponent; e.g. 3 with exponent denotes the limit of the sequence

Three important limits.

Theorem 1. If θ is in radians, then

Theorem 2. If θ is in radians, then

Theorem 3. The number e is given by

Logarithms to the base e are called natural logarithms. Loge x is often called log x (base omitted) or ln x.

Def. Exponential function. (1) The function ex.

(2) The function ax, where a is a positive constant. If a 1, the function ax is the inverse of the logarithmic function logax.

James & James. Mathematics Dictionary.

The exponential function y = ax is shown in Fig. 1. Its derivative is everywhere positive and increases with increasing values of x.

Def. Logarithmic function y = loga x (a > 1). The logarithmic function is defined as the inverse of the exponential function i.e. if

x = ay

then

y = loga x .

See Fig. 2. Its graph is the same as that of y = ax with the axes interchanged. It is defined only for positive values of x.

The derivative of loga u.

Theorem.

Proof. We wish to find the derivative of the function

y = loga x .

We proceed by applying the fundamental differentiation process. Starting at any point P on the curve and letting x increase by an amount Δx we have

Let us now multiply numerator and denominator by x giving

If now Δx 0, the quantity

approaches the number e because it is of the form

with v approaching zero. We have then

End of proof.

For the special case of natural logarithms where a = e this becomes

If we consider the function y = loga u where u is a differentiable function of x, its derivative with respect to x is given by

For the special case of natural logarithms this becomes

Example. Find dy/dx for

Solution.

The derivative of au.

Theorem.

Proof. If

y = ax

then

x = loga y .

Taking the derivative with respect to y

or

= y loge a

= ax loge a

End of proof.

If u is a differentiable function of x we have

If a is the number e, we have

Example. Find dy/dx for

Solution.

Logarithmic differentiation. In obtaining the derivatives of certain kinds of functions, such as complicated products and quotients, and functions of the type uv, where u and v are both variable, it is often easier to take the logarithm before differentiating. This process is called logarithmic differentiation.

Example 1. Find dy/dx if

Solution. Taking logarithms of both sides

ln y = l ln u + m ln v - n ln w

Differentiating with respect to x

Solving for dy/dx

Example 2. Find dy/dx if

Solution. Taking logarithms of both sides

Differentiating with respect to x

Solving for dy/dx

The derivative of uv. The derivative of a function of the form

y = uv,

where u and v are differentiable functions of x, can be found by the method of logarithmic differentiation.

Example. Differentiate y = (x + 1)x.

Solution. Taking the natural logarithm of each side, we obtain

ln y = x ln (x + 1).

Now differentiating this relation implicitly with respect to x we have

Solving for dy/dx we get

Note that neither the formula for the derivative of un nor the derivative for av applies to functions of this type.

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