For a > 0 an x any real number, we efine Lecture 4 : General Logarithms an Exponentials. a x = e x ln a, a > 0. The function a x is calle the exponential function with base a. Note that ln(a x ) = x ln a is true for all real numbers x an all a > 0. (We saw this before for x a rational number). Note: We have no efinition for a x when a < 0, when x is irrational. For example 2 2 = e 2 ln 2, 2 2, ( 2) 2 (no efinition). Algebraic rules The following Laws of Exponent follow from the laws of exponents for the natural exponential function. a x+y = a x a y a x y = ax (a x ) y = a xy (ab) x = a x b x a y Proof a x+y = e (x+y) ln a = e x ln a+y ln a = e x ln a e y ln a = a x a y. etc... Example Simplify (ax ) 2 a x2 +1 a 2. Differentiation The following ifferentiation rules also follow from the rules of ifferentiation for the natural exponential. x (ax ) = x (ex ln a ) = a x ln a x (ag(x) ) = x eg(x) ln a = g (x)a g(x) ln a Example Differentiate the following function: f(x) = (1000)2 x2 +1. Graphs of Exponential functions. Case 1: 0 < a < 1 y-intercept: The y-intercept is given by y = a 0 = e 0 ln a = e 0 = 1. x-intercept: The values of a x = e x ln a are always positive an there is no x intercept. 1
Slope: If 0 < a < 1, the graph of y = a x has a negative slope an is always ecreasing, x (ax ) = a x ln a < 0. In this case a smaller value of a gives a steeper curve. The graph is concave up since the secon erivative is 2 x 2 (a x ) = a x (ln a) 2 > 0. As x, x ln a approaches, since ln a < 0 an therefore a x = e x ln a 0. As x, x ln a approaches, since both x an ln a are less than 0. Therefore a x = e x ln a. For 0 < a < 1, lim a x = 0, x lim x ax =. y 1 2 x 0 y 1 4 x 40 y 1 8 x 30 y 1 x 20 10 4 2 2 4 Graphs of Exponential functions. Case 2: a > 1 y-intercept: The y-intercept is given by y = a 0 = e 0 ln a = e 0 = 1. x-intercept: The values of a x = e x ln a are always positive an there is no x intercept. If a > 1, the graph of y = a x has a positive slope an is always increasing, x (ax ) = a x ln a > 0. The graph is concave up since the secon erivative is 2 x 2 (a x ) = a x (ln a) 2 > 0. In this case a larger value of a gives a steeper curve. As x, x ln a approaches, since ln a > 0 an therefore a x = e x ln a As x, x ln a approaches, since x < 0 an ln a > 0. Therefore a x = e x ln a 0. For a > 1, lim a x =, x lim x ax = 0. y 2 x y 4 x 120 100 y 8 x 80 60 40 20 4 2 2 4 2
Functions of the form (f(x)) g(x). Derivatives We now have 4 ifferent types of functions involving bases an powers. So far we have ealt with the first three types: If a an b are constants an g(x) > 0 an f(x) an g(x) are both ifferentiable functions. x ab = 0, x (f(x))b = b(f(x)) b 1 f (x), x ag(x) = g (x)a g(x) ln a, x (f(x))g(x) For x (f(x))g(x), we use logarithmic ifferentiation or write the function as (f(x)) g(x) = e an use the chain rule. Example Differentiate x 2x2, x > 0. g(x) ln(f(x)) Limits To calculate limits of functions of this type it may help write the function as (f(x)) g(x) = e g(x) ln(f(x)). Example What is lim x x x General Logarithmic functions Since f(x) = a x is a monotonic function whenever a 1, it has an inverse which we enote by f 1 (x) = log a x. We get the following from the properties of inverse functions: f 1 (x) = y if an only if f(y) = x log a (x) = y if an only if a y = x f(f 1 (x)) = x f 1 (f(x)) = x a log a (x) = x log a (a x ) = x. 3
Converting to the natural logarithm It is not ifficult to show that log a x has similar properties to ln x = log e x. This follows from the Change of Base Formula which shows that The function log a x is a constant multiple of ln x. log a x = ln x ln a The algebraic properties of the natural logarithm thus exten to general logarithms, by the change of base formula. log a 1 = 0, log a (xy) = log a (x) + log a (y), log a (x r ) = r log a (x). for any positive number a 1. In fact for most calculations (especially limits, erivatives an integrals) it is avisable to convert log a x to natural logarithms. The most commonly use logarithm functions are log 10 x an ln x = log e x. Since log a x is the inverse function of a x, it is easy to erive the properties of its graph from the graph y = a x, or alternatively, from the change of base formula log a x = ln x ln a. 10 10 4 2 2 4 y 2 x y 1 2 x y Log 2 x y Log 1 2 x 4 2 2 4 Basic Application Example Express as a single number log 2 log 4
Using the change of base formula for Derivatives From the above change of base formula for log a x, we can easily erive the following ifferentiation formulas: x (log a x) = 1 x ln a x (log a g(x)) = g (x) g(x) ln a. Example Fin x log 2(x sin x). A special limit an an approximation of e We erive the following limit formula by taking the erivative of f(x) = ln x at x = 1: ln(1 + x) lim x 0 x = lim x 0 ln(1 + x) 1/x = 1. Applying the (continuous) exponential function to the limit we get e = lim x 0 (1 + x) 1/x Note If we substitute y = 1/x in the above limit we get e = lim (1 + 1 ) y an e = lim (1 + 1 ) n y y n n where n is an integer (see graphs below). We look at large values of n below to get an approximation of the value of( e. ) n ( n n = 10 1 + 1 n = 2.9374246, n = 100 1 + n) 1 = 2.70481383, n = 100 ( ) n ( n 1 + 1 n = 2.71692393, n = 1000 1 + n) 1 = 2.181493. Example Fin lim x 0 (1 + x 2 )1/x.
9 8 7 6 4 3 1.0 0. 0. 1.0 2.70 2.68 2.66 2.64 2.62 2.60 2.8 2.6 20 40 60 80 100 2.70 2.68 2.66 2.64 2.62 2.60 2.8 20 40 60 80 100 6