2.6 Logarithmic Functions. Inverse Functions. Question: What is the relationship between f(x) = x 2 and g(x) = x?

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1 Inverse Functions Question: What is the relationship between f(x) = x 3 and g(x) = 3 x? Question: What is the relationship between f(x) = x 2 and g(x) = x?

2 Definition (One-to-One Function) A function f is said to be one-to-one if each range value corresponds to exactly one domain value. Definition (Inverse of a Function) If f is a one-to-one function, then the inverse of f is the function formed by interchanging the independent and dependent variables for f. Thus, if (a, b) is a point on the graph of f, then (b, a) is a point on the graph of the inverse of f. Note: If f is not one-to-one, then f does not have an inverse. So g(x) = 3 x is the inverse of f(x) = x 3, and vice versa. However, h(x) = x 2 does not have an inverse, since it is not one-to one. Another examples are 2

3 Logarithmic Functions If we start with the exponential function f defined by y = 2 x and interchange the variables, we obtain the inverse of f: x = 2 y We call the inverse the logarithmic function with base 2, and write y = log 2 x if and only if x = 2 y. We can graph y = log 2 x by graphing x = 2 y since they are equivalent. Any ordered pair of numbers on the graph of the exponential function will be on the graph of the logarithmic function if we interchange the order of the components. x y = 2 x x = 2 y y y 9 y = 2 x 7 y = x 5 3 x = 2 y x 3

4 Definition (Logarithmic Function) The inverse of an exponential function is called a logarithmic function. For b > 0 and b 1, y = log b x is equivalent to x = b y The log to the base b of x is the exponent to which b must be raised to obtain x. Domain and Range of Logarithmic Function The domain of the logarithmic function is the set of all positive real numbers, which is also the range of the corresponding exponential function; and the range of the logarithmic function is the set of all real numbers, which is also the domain of the corresponding exponential function. y y 9 7 y = b x y = x y = b x 9 7 y = x 5 3 x = b y x x x = b y 4

5 Example 1 Change each logarithmic form to an equivalent exponential form: (a) log 3 9 = 2 (b) log 4 2 = 1 2 (c) log = 2 Example 2 Change each exponential form to an equivalent logarithmic form: (a) 49 = 7 2 (b) 6 = 36 (c) 1 3 = 3 1 Example 3 Find y, b, or x, as indicated (a) Find y: y = log 9 27 (b) Find x: log 3 x = 1 (c) Find b: log b 1000 = 3 5

6 Properties of Logarithmic Functions Recall one of the properties of an exponential function b x b y = b x+y. Let M = b x, N = b y. Then So log b MN = log b M + log b N. Similarly we can derive some other properties Theorem 1 (Properties of Logarithmic Function) If b, M, and N are positive real numbers, b 1, and p and x are real numbers, then 1. log b 1 = 0 2. log b b = 1 3. log b b x = x 4. b log b x = x, x > 0 5. log b MN = log b M + log b N 6. log b M N = log b M log b N 7. log b M p = p log b M 8. log b M = log b N if and only if M = N 6

7 Example 4 (a) log b x yz = (b) log a ( w v ) 3 2 = (c) 2 u log 2 b = (d) log 2 x log 2 b = 7

8 Example 5 Solve for x: (a) 3 log b log b 25 log b 20 = log b x (b) log 3 x + log 3 (x 3) = log

9 Common Logarithms and Natural Logarithms Common Logarithms Common logarithms are logarithms with base 10. denoted as log x = log 10 x They usually Natural Logarithms Natural logarithms are logarithms with base e. They usually denoted as ln x = log e x 9

10 Applications Example 6 (Doubling Time for an Investment) How long will it take money to double if it is invested at 13% compounded annually? 10