Slide Copyright © 2012 Pearson Education, Inc.

9.3 Logarithmic Functions ■ Graphs of Logarithmic Functions ■ Common Logarithms ■ Equivalent Equations ■ Solving Certain Logarithmic Equations

Slide 9- 3 Copyright © 2012 Pearson Education, Inc. Consider the exponential function f (x) = 3 x. Like all exponential functions, f is one-to-one. Can a formula for f -1 be found? To answer this, we use the method of Section 12.1: f -1 (x) = the exponent to which we must raise 3 to get x. y = 3 x x = 3 y y = the exponent to which we must raise 3 to get x.

Slide 9- 4 Copyright © 2012 Pearson Education, Inc. We now define a new symbol to replace the words “the exponent to which we must raise 3 to get x”: log 3 x, read “the logarithm, base 3, of x,” or “log, base 3, of x,” means “the exponent to which we raise 3 to get x.” Thus if f (x) = 3 x, then f -1 (x) = log 3 x. Note that f -1 (9) = log 3 9 = 2, because 2 is the exponent to which we raise 3 to get 9.

Slide 9- 5 Copyright © 2012 Pearson Education, Inc. Example Solution a) log 3 81 b) log 3 1 c) log 3 (1/9) a) Think of log 3 81 as the exponent to which we raise 3 to get 81. That exponent is 4. Therefore, log 3 81 = 4. Simplify: b) We ask: “To what exponent do we raise 3 in order to get 1?” That exponent is 0. Thus, log 3 1 = 0. c) To what exponent do we raise 3 in order to get 1/9? Since 3 -2 = 1/9, we have log 3 (1/9) = –2.

Slide 9- 6 Copyright © 2012 Pearson Education, Inc. Graphs of Logarithmic Functions For any exponential function f (x) = a x, the inverse is called a logarithmic function, base a. The graph of the inverse can be drawn by reflecting the graph of f (x) = a x across the line y = x. It will be helpful to remember that the inverse of f (x) = a x is given by f –1 (x) = log a x.

Slide 9- 7 Copyright © 2012 Pearson Education, Inc. The Meaning of log a x For x > 0 and a a positive constant other than 1, log a x is the exponent to which a must be raised in order to get x. Thus, log a x = m means a m = x or equivalently, log a x is that unique exponent for which

Slide 9- 8 Copyright © 2012 Pearson Education, Inc. Example Solution Simplify: Remember that log 5 23 is the exponent to which 5 is raised to get 23. Raising 5 to that exponent, we have

Slide 9- 9 Copyright © 2012 Pearson Education, Inc. It is important to remember that a logarithm is an exponent.

Slide Copyright © 2012 Pearson Education, Inc. Example Graph y = f (x) = log 3 x. Solution y 1 3 1/3 9 1/ –1 2 –2 3

Slide Copyright © 2012 Pearson Education, Inc. Common Logarithms Base-10 logarithms, called common logarithms, are useful because they have the same base as our “commonly” used decimal system.

Slide Copyright © 2012 Pearson Education, Inc. Example Solution Graph: log x/4 – 2.

Slide Copyright © 2012 Pearson Education, Inc. Equivalent Equations We use the definition of logarithm to rewrite a logarithmic equation as an equivalent exponential equation or the other way around: m = log a x is equivalent to a m = x.

Slide Copyright © 2012 Pearson Education, Inc. Example Solution exponential equation: a) –m = log 3 x b) 6 = log a z Rewrite each as an equivalent a) –m = log 3 x is equivalent to 3 -m = x b) 6 = log a z is equivalent to a 6 = z. The base remains the base. The logarithm is the exponent.

Slide Copyright © 2012 Pearson Education, Inc. Example Solution logarithmic equation: a) 49 = 7 x b) x -2 = 9 Rewrite each as an equivalent a) 49 = 7 x is equivalent to x = log 7 49 b) x -2 = 9 is equivalent to –2 = log x 9. The base remains the base. The exponent is the logarithm.

Slide Copyright © 2012 Pearson Education, Inc. Solving Certain Logarithmic Equations Logarithmic equations are often solved by rewriting them as equivalent exponential equations.

Slide Copyright © 2012 Pearson Education, Inc. Example Solution Solve: a) log 3 x = –3; b) log x 4 = 2. a) log 3 x = –3 x = 3 –3 = 1/27 b) log x 4 = 2 4 = x 2 x = 2 or x = –2 Because all logarithmic bases must be positive, –2 cannot be a solution. The solution is 2. The solution is 1/27. The check is left to the student.

Slide Copyright © 2012 Pearson Education, Inc. The Principle of Exponential Equality For any real number b, where (Powers of the same base are equal if and only if the exponents are equal.)

Slide Copyright © 2012 Pearson Education, Inc. Example Solution Solve: a) log 6 36 = x; b) log 9 1 = t. a) log 6 36 = x 6 x = 36 x = 2 The solution to part a) is 2. The solution to part b) is 0. The check is left to the student. 6 x = 6 2 b) log 9 1 = t 9 t = 1 9 t = 9 0 t = 0

Slide Copyright © 2012 Pearson Education, Inc. log a 1 The logarithm, base a, of 1 is always 0: log a 1 = 0. log a a The logarithm, base a, of a is always 1: log a a = 1.