Chapter 6 | Applications of Integration
729
6.39 Evaluate the following integral: ∫ 4 e 3 x dx .
General Logarithmic and Exponential Functions We close this section by looking at exponential functions and logarithms with bases other than e . Exponential functions are functions of the form f ( x ) = a x . Note that unless a = e , we still do not have a mathematically rigorous definition of these functions for irrational exponents. Let’s rectify that here by defining the function f ( x ) = a x in terms of the exponential function e x . We then examine logarithms with bases other than e as inverse functions of exponential functions.
Definition For any a >0, and for any real number x , define y = a x as follows: y = a x = e x ln a .
Now a x is defined rigorously for all values of x . This definition also allows us to generalize property iv. of logarithms and property iii. of exponential functions to apply to both rational and irrational values of r . It is straightforward to show that
properties of exponents hold for general exponential functions defined in this way. Let’s now apply this definition to calculate a differentiation formula for a x . We have d dx a x = d dx e x ln a = e x ln a ln a = a x ln a . The corresponding integration formula follows immediately.
Theorem 6.20: Derivatives and Integrals Involving General Exponential Functions Let a >0. Then, d dx a x = a x ln a and ∫ a x dx = 1 ln a a x + C .
If a ≠1, then the function a x is one-to-one and has a well-defined inverse. Its inverse is denoted by log a x . Then, y = log a x if and only if x = a y . Note that general logarithm functions can be written in terms of the natural logarithm. Let y = log a x . Then, x = a y . Taking the natural logarithm of both sides of this second equation, we get
ln x = ln( a y ) ln x = y ln a y = ln x ln a
log x = ln x ln a . Thus, we see that all logarithmic functions are constant multiples of one another. Next, we use this formula to find a differentiation formula for a logarithm with base a . Again, let y = log a x . Then,
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