7 INVERSE FUNCTIONS INVERSE FUNCTIONS 7 4 General

7 INVERSE FUNCTIONS

INVERSE FUNCTIONS 7. 4* General Logarithmic and Exponential Functions In this section, we: Use the natural exponential and logarithmic functions to study exponential and logarithmic functions with base a > 0.

GENERAL EXPONENTIAL FUNCTIONS If a > 0 and r is any rational number, then, by Equations 4 and 7 in Section 7. 3*, ar = (eln a)r = er ln a

GENERAL EXP. FUNCTIONS Definition 1 Thus, even for irrational numbers x, we define: ax = ex ln a § Thus, for instance,

GENERAL EXP. FUNCTIONS The function f(x) = ax is called the exponential function with base a. § Notice that ax is positive for all x because ex is positive for all x.

GENERAL EXP. FUNCTIONS Definition 1 allows us to extend one of the laws of logarithms. § We know that ln(ar) = r ln a when r is rational. § However, if we now let r be any real number we have, from Definition 1, ln ar = ln(er ln a) = r ln a

GENERAL EXP. FUNCTIONS Hence, we have: ln ar = r ln a for any real number r. Equation 2

GENERAL EXP. FUNCTIONS The general laws of exponents follow from Definition 1 together with the laws of exponents for ex.

LAWS OF EXPONENTS Laws 3 If x and y are real numbers and a, b > 0, then 1. ax+y = axay 2. ax-y = ax/ay 3. (ax)y = axy 4. (ab)x = axbx

LAW 1 OF EXPONENTS Proof Using Definition 1 and the laws of exponents for ex, we have:

LAW 3 OF EXPONENTS Proof Using Equation 2, we obtain: § The remaining proofs are left as exercises.

GENERAL EXP. FUNCTIONS Formula 4 The differentiation formula for exponential functions is also a consequence of Definition 1:

GENERAL EXP. FUNCTIONS Formula 4—Proof

GENERAL EXP. FUNCTIONS Notice that, if a = e, then ln e = 1 and Formula 4 simplifies to a formula we already know: (d/dx) ex = ex § In fact, the reason the natural exponential function is used more often than other exponential functions is that its differentiation formula is simpler.

GENERAL EXP. FUNCTIONS Example 1 In Example 6 in Section 3. 7, we considered a population of bacteria cells in a homogeneous nutrient medium. § We showed that, if the population doubles every hour, then the population after t hours is: n = n 02 t where n 0 is the initial population.

GENERAL EXP. FUNCTIONS Example 1 Now, we can use Formula 4 to compute the growth rate: § For instance, if the initial population is n 0 = 1000 cells, then the growth rate after two hours is:

GENERAL EXP. FUNCTIONS Example 2 Combining Formula 4 with the Chain Rule, we have:

EXPONENTIAL GRAPHS If a > 1, then ln a > 0. So, (d/dx) ax = ax ln a > 0. § This shows that y = ax is increasing. © Thomson Higher Education

EXPONENTIAL GRAPHS If 0 < a < 1, then ln a < 0. § So, y = ax is decreasing. © Thomson Higher Education

EXPONENTIAL GRAPHS Notice from this figure that, as the base a gets larger, the exponential function grows more rapidly (for x > 0). © Thomson Higher Education

EXPONENTIAL GRAPHS Here, we see how the exponential function y = 2 x compares with the power function y = x 2. © Thomson Higher Education

EXPONENTIAL GRAPHS The graphs intersect three times. § Ultimately, though, the exponential curve y = 2 x grows far more rapidly than the parabola y = x 2. © Thomson Higher Education

EXPONENTIAL GRAPHS In Section 7. 5, we will show exponential functions occur in the description of population growth and radioactive decay. § Let’s look at human population growth.

EXPONENTIAL GRAPHS The table shows data for the population of the world in the 20 th century. The figure shows the related scatter plot. © Thomson Higher Education

EXPONENTIAL GRAPHS The pattern of the data points in the figure suggests exponential growth. © Thomson Higher Education

EXPONENTIAL GRAPHS Hence, we use a graphing calculator with exponential regression capability to apply the method of least squares and obtain the exponential model P = (0. 008079266). (1. 013731)t

EXPONENTIAL GRAPHS The figure shows the graph of the exponential function together with the original data points. § We see the curve fits the data reasonably well. § The period of relatively slow population growth is explained by the two world wars and the Great Depression of the 1930 s. © Thomson Higher Education

EXPONENTIAL INTEGRALS The integration formula that follows from Formula 4 is:

EXPONENTIAL INTEGRALS Example 3

THE POWER RULE VS. THE EXPONENTIAL RULE Now that we have defined arbitrary powers of numbers, we are in a position to prove the general version of the Power Rule—as promised in Section 3. 3

THE POWER RULE If n is any real number and f(x) = xn, then f ’(x) = n-1 nx

THE POWER RULE Proof Let y = xn and use logarithmic differentiation: ln | y | = ln | x |n = n ln | x | § Therefore, § Hence, x≠ 0

POWER RULE VS. EXP. RULE Note You should distinguish carefully between: § The Power Rule [(d/dx) xn = nxn-1], where the base is variable and the exponent is constant. § The rule for differentiating exponential functions [(d/dx) ax = ax ln a], where the base is constant and the exponent is variable.

THE POWER RULE—CASES In general, there are four cases for exponents and bases:

THE POWER RULE E. g. 4—Solution 1 Differentiate: § Using logarithmic differentiation, we have:

THE POWER RULE E. g. 4—Solution 2 Another method is to write

THE POWER RULE The figure illustrates Example 4 by showing the graphs of its derivative. © Thomson Higher Education and

GENERAL LOG. FUNCTIONS If a > 0 and a ≠ 1, then f(x) = ax is a one-to-one function. § Its inverse function is called the logarithmic function with base a. § It is denoted by loga.

GENERAL LOG. FUNCTIONS Formula 5 Thus, loga x = y ay = x § In particular, we see that: logex = ln x

GENERAL LOG. FUNCTIONS The cancellation equations for the inverse functions logax and ax are:

GENERAL LOG. FUNCTIONS The figure shows the case where a > 1. § The most important logarithmic functions have base a > 1. § The fact that y = ax is a very rapidly increasing function for x > 0 is reflected in the fact that y = logax is a very slowly increasing function for x > 1. © Thomson Higher Education

GENERAL LOG. FUNCTIONS The figure shows the graphs of y = logax with various values of the base a. § Since loga 1 = 0, the graphs of all logarithmic functions pass through the point (1, 0). © Thomson Higher Education

GENERAL LOG. FUNCTIONS The laws of logarithms are similar to those for the natural logarithm. § They can be deduced from the laws of exponents. § See Exercise 65.

CHANGE OF BASE FORMULA Formula 6 This formula shows that logarithms with any base can be expressed in terms of the natural logarithm. § For any positive number a (a ≠ 1), we have:

CHANGE OF BASE FORMULA Formula 6—Proof Let y = logax. Then, from Formula 5, we have: ay = x § Taking natural logarithms of both sides of this equation, we get: y ln a = ln x § Therefore,

CHANGE OF BASE FORMULA Scientific calculators have a key for natural logarithms. So, Formula 6 enables us to: § Use a calculator to compute a logarithm with any base (as shown in the following example). § Graph any logarithmic function on a graphing calculator or computer (as in Exercises 14 -16).

CHANGE OF BASE FORMULA Example 5 Evaluate log 85 correct to six decimal places. § Formula 6 gives:

CHANGE OF BASE FORMULA Formula 7 Formula 6 enables us to differentiate any logarithmic function. § Since ln a is a constant, we can differentiate as follows:

CHANGE OF BASE FORMULA Example 6 Using Formula 7 and the Chain Rule, we get:

CHANGE OF BASE FORMULA From Formula 7, we see one of the main reasons that natural logarithms (logarithms with base e) are used in calculus: § The differentiation formula is simplest when a = e because ln e = 1.

THE NUMBER e AS A LIMIT We have shown that, if f(x) = ln x, then f’(x) = 1/x. Thus, f’(1) = 1. § We now use this fact to express the number e as a limit.

THE NUMBER e AS A LIMIT From the definition of a derivative as a limit, we have:

THE NUMBER e AS A LIMIT Since f’(1) = 1, we have: § Then, by Theorem 8 in Section 2. 5 and the continuity of the exponential function, we have:

THE NUMBER e AS A LIMIT Formula 8 The number e as a limit:

THE NUMBER e AS A LIMIT Formula 8 is illustrated by: § The graph of the function y = (1 + x)1/x. § A table of values for small values of x. © Thomson Higher Education/

THE NUMBER e AS A LIMIT Formula 9 If we put n = 1/x in Formula 8, then n → ∞ as x → 0+. § So, an alternative expression for e is: