This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Ch5/MATH1804/YMC/200910 1 Chapter 5. Exponential and Logarithmic Functions 5.1. Review • Exponential Functions An exponential function is a function of the form f ( x ) = a x where a > , a 6 = 1 . Note that the exponent x can be any real number here, and the number a , called the base , is a constant. There is a very important exponential function that arises naturally in many places. This function is called the natural exponential function whose base is given by the number e . • Natural Domain and Range of a x The natural domain for exponential functions are R and the ranges are (0 , ∞ ) . • Laws of Exponents 1 . a x · a y = a x + y 5 . a b x = a x b x 2 . ( a x ) y = a xy 6 . a 1 = a 3 . ( ab ) x = a x · b x 7 . a = 1 4 . a x a y = a x y 8 . a x = 1 a x • Derivatives of Exponential Functions Note that d dx a x = lim h → a x + h a x h = lim h → a x a h a x h = a x lim h → a h 1 h . The number e can be defined by the equation d dx e x = e x . • Logarithmic Functions Since the exponential functions are injective, it follows that each exponential function has an inverse. These functions are called logarithmic functions . We denote by log b x the logarithmic function with base b . • Natural Domain and Range of log b x The natural domain for logarithmic functions are (0 , ∞ ) and the ranges are R . Ch5/MATH1804/YMC/200910 2 • For any x ∈ (0 , ∞ ) , we have y = log b x is equivalent to b y = x. It follows that log b b x = x for any x ∈ R and b log b x = x for any x ∈ (0 , ∞ ) . (5.1) • Common and Natural Logarithm Logarithm to the base 10 is called common logarithm . Important to Calculus is the logarithm to the base e , the natural logarithm . We use the notation ln to mean log e . In this case, (5.1) becomes ln e x = x for any x ∈ R and e ln x = x for any x ∈ (0 , ∞ ) . (5.2) • The graph of e x and ln x is shown below: • Properties of Logarithmic Functions 1. log b ( xy ) = log b x + log b y 2. log b x y = log b x log b y 3. log b ( x r ) = r log b x 4. log b x = log a x log a b (ChangeofBase Formula) In particular, if we take a to be the number e , then log b x = ln x ln b (5.3) Ch5/MATH1804/YMC/200910 3 5.2. Derivatives of Exponential and Logarithmic Functions As we have discussed, d dx e x = e x ....
View
Full
Document
This note was uploaded on 11/07/2010 for the course SCIENCE math1804 taught by Professor Prof during the Spring '10 term at HKU.
 Spring '10
 Prof

Click to edit the document details