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5.1 & 5.2 The Natural Logarithmic Function and Calculus
Recall that the Power Rule has an important disclaimer: x n 1 x n dx C, n 1 n 1 1 We must find an antiderivative for the function f (x) . x We define this function in a new class of function called logarithmic functions. This particular function is the natural logarithmic function.
Definition of the Natural Logarithmic Function
t 1 The natural logarithmic function is defined by ln x dt, x 0 1 t 1 (a) Graph of y (b) Graph of y ln x t
The domain of the natural logarithmic function is the set of all positive real numbers.
From the definition , we can see that ln x is ______for x 1 and ln x is ______for 0 x 1, and ln(1) _____ when x 1.
Theorem – Properties of the Natural Logarithmic Function The natural logarithmic function has the following properties: 1. The domain is 0, and the range is , . 2. The function is continuous, increasing, and one-to-one. 3. The graph is concave downward.
Logarithmic Properties If a and b are positive numbers and n is rational, then the following properties are true: 1. ln(1) 0 2. lnab lna lnb 3. lnan nlna a 4. ln lna lnb b
If you completed your log packet then you should remember how to use these properties!!
The Number e In logarithms you have studied so far, the logs have been defined with a base - usually base 10. For example, log10 10 1. To define the base for the natural logarithm, we use the properties. There must be a real number x such that ln x 1. This number is denoted by the letter e . e is irrational and has the decimal approximation: e 2.71828182846
Also, loge 1 and lne 1.
Definition of e e 1 The letter e denotes the positive real number such that: lne dt 1 1 t You can now use logarithmic properties to evaluate the natural logarithms of several other numbers. For example, ln e n =
Using this, we can evaluate ln e n for various powers of n , as shown in the table:
x 1 1 1 e0 e e2
e3 e 2 e ln x
Example #1 Evaluate each of the following: a. ln 2 b. ln32 c. ln0.1
Theorem –Derivative of the Natural Logarithmic Function Let u be a differentiable function of x . d 1 d 1 du u ' Theorem 1 [ln x] , x 0 Theorem 2 [lnu] , u 0 dx x dx u dx u
Example #2 Differentiate each of the following logarithmic functions: d d a. ln(2x) b. ln(x 2 1) dx dx
d d c. x ln x d. ln x 3 dx dx
Napier used logarithmic properties to simplify calculations. With calculators this is no longer necessary, but we still use the properties for differentiation. 2 x x 2 1 Example #3 Differentiate f (x) ln x 1 Example #4 Differentiate f (x) ln 2x3 1
On occasion it is convenient to use logarithms as aids in differentiating non-logarithmic functions. This procedure is called logarithmic differentiation.
(x 2) 2 Example #5 Find the derivative of y , x 2 x 2 2
Because the natural logarithm is undefined for negative numbers, you will often encounter expressions of the form lnu . When you differentiate functions in the form y lnu do so as if the absolute value were not present.
Theorem – Derivative Involving Absolute Value d u ' If u is a differentiable function of x such that u 0 , then lnu dx u
Example #6 Find the derivative of f (x) ln cosx
Example #7 Locate the relative extrema of y ln(x 2 2x 3) . Justify your response.
Integration! Theorem: Log Rule for Integration Let u be a differentiable function of x . 1 1 1. dxln x C 2. duln u C x u u' Since du u' dx , the second formula can also be written as dxln u C u 2 1 Example #1 Evaluate: dx Example #2 Evaluate: dx x 41x
x Example #3 Find the area of the region bounded by the graph of y , the x axis, and the line x 3. x2 1
Recognizing Quotient Forms of the Log Rule Example #4 Evaluate each of the following: 31x2 sec2 x a. dx b. dx xx3 tan x
x 1 1 c. d. dx xx2 2 32x
Note: With antiderivatives involving logarithms, it is easy to obtain forms that look quite different but are still equivalent. Which of the following are equivalent to the antiderivative in Ex. 4d? 1 12 1 ln (3xC 2)3 ln xC ln (3xC 2)3 33
Integrals to which the Log Rule can be applied often appear in disguised form. For instance, if a rational function has a numerator of degree greater than or equal to that of the denominator, division may reveal a form to which you can apply the Log Rule. xx2 1 Example #5 Evaluate dx (Hint: Use long division!) x2 1
2x Example #6 Evaluate dx (Hint: use change of variables) (x 1)2
As we continue the study of integration, we will devote much time to integration techniques. To master these techniques you must recognize the “form-fitting” nature of integration. In this sense, integration is not nearly as straightforward as differentiation. So, be ready to THINK until the lightbulb goes off!
Guidelines for Integration: 1. Memorize a basic list of integration formulas (12 total so far: Power Rule, Log Rule, and ten trig rules). 2. Find an integration formula that resembles all or part of the integrand, and, by trial and error, find a choice of u that will make the integrand conform to the formula. 3. If you cannot find a u substitution that works, try altering the integrand. You might try a trig identity, multiplication and division by the same quantity, or addition and subtraction of the same quantity. Be creative!
dy 1 Example #7 Solve the differential equation dx xln x
Example #8 Evaluate tan xdx Example #9 Evaluate sec xdx
Integrals for the Six Basic Trigonometric Function: sinudu cos u C cos udu sin u C
tanudu ln cos u C cot udu ln sin u C
secudu ln sec u tan u C csc udu ln csc u cot u C
4 Example #10 Evaluate: 1 tan2 xdx 0
Example #11 The electromotive force E of a particular electrical circuit is given by Et3sin2 where E is measured in volts and t is measured in seconds. Find the average value of E as t ranges from 0 to 0.5 seconds.