THE NATURE OF GROWTH Copyright © Cengage Learning. All rights reserved. 10

Copyright © Cengage Learning. All rights reserved Exponential Equations

3 The measurement of growth and decay often involves the study of relatively large or relatively small quantities. Difficulty with scaling measurements is often one of our primary concerns when describing and measuring figures and data. Large and small numbers are often represented in exponential form and scientific notation.

4 Exponential Equations In this section, we investigate solving equations known as exponential equations An exponential is an expression of the form b x ; The number b is called the base. In definition b x is defined as an integer.

5 Exponential Equations Let’s solve the exponential equation 2 x = 14. To solve an equation means to find the replacement(s) for the variable that make the equation true. You might try certain values: x = 1: 2 x = 2 1 = 2 Too small x = 2: 2 x = 2 2 = 4 Too small x = 3: 2 x = 2 3 = 8 Still too small x = 4: 2 x = 2 4 = 16 Too big It seems as if the number you are looking for is between 3 and 4. Our task in this section is to find both an approximate as well as an exact value for x.

6 Definition of Logarithm

7 The solution of the equation 2 x = 14 seeks an x-value. What is this x-value? We express the idea in words: x is the exponent on a base 2 that gives the answer 14 This can be abbreviated as x = exp on base 2 to give 14

8 Definition of Logarithm We further shorten this notation to x = exp 2 14 This statement is read, “x is the exponent on a base 2 that gives the answer 14.” It appears that the equation is now solved for x, but this is simply a notational change.

9 Definition of Logarithm The expression “exponent of 14 to the base 2” is called, for historical reasons, “the log of 14 to the base 2.” That is, x = exp 2 14 and x = log 2 14 mean exactly the same thing.

10 Definition of Logarithm This leads us to the following definition of logarithm. The statement x = log b A should be read as “x is the log (exponent) on a base b that gives the value A.” Do not forget that a logarithm is an exponent.

11 Example 2 – Write exponentials in logarithmic form Write in logarithmic form: a. 5 2 = 25 Solution: a. In 5 2 = 25, 5 is the base and 2 is the exponent, so we write 2 = log 5 25 Remember, the logarithmic expression “solves” for the exponent.

12 Example 2 – Solution b. With = 2 –3, the base is 2 and the exponent is –3: –3 = log 2 c. With = 8, the base is 64 and the exponent is (since = 64 1/2 ): = log 64 8 cont’d

13 Definition of Logarithm In elementary work, the most commonly used base is 10, so we call a logarithm to the base 10 a common logarithm, and we agree to write it without using a subscript 10. That is, log x is a common logarithm. A logarithm to the base e is called a natural logarithm and is denoted by ln x. The expression ln x is often pronounced “ell en x” or “lon x.”

14 Definition of Logarithm The solution for the equation 10 x = 2 is x = log 2, and the solution for the equation e x = 0.56 is x = ln 0.56.

15 Evaluating Logarithms

16 Evaluating Logarithms To evaluate a logarithm means to find a numerical value for the given logarithm. To evaluate a logarithm to some base other than base 10 or base e. The first method uses the definition of logarithm that is, For positive b and A, b ≠ 1, x = log b A means b x = A. x is called the logarithm and A is called the argument. The second method uses what is called the change of base theorem. Before we state this theorem, we consider its plausibility with the next example.

17 Example 6 – Evaluate logarithmic expressions Evaluate the given expression. Solution: a. From the definition of logarithm, log 2 8 = x means 2 x = 8 or x = 3. Thus, log 2 8 = 3. By calculator, Also,

18 Evaluating Logarithms

19 Example 7 – Evaluate logarithms with a change of base Evaluate (round to the nearest hundredth): a. log 7 3 b. log Solution:

20 Evaluating Logarithms We now return to the problem of solving 2 x = 14 x = log 2 14 We call log 2 14 the exact solution for the equation, and Example 8 finds an approximate solution. Given equation. Solution

21 Example 8 – Solve an exponential equation Solve 2 x = 14 (correct to the nearest hundredth). Solution: We use the definition of logarithm and the change of base theorem to write

22 Exponential Equations

23 Exponential Equations We now turn to solving exponential equations. Exponential equations will fall into one of three types: The next example illustrates the procedure for solving each type of exponential equation.

24 Example 9 – Solve exponential equations with common and natural logs Solve the following exponential equations: a. 10 x = 5 b. e –0.06x = c. 8 x = Solution: Regardless of the base, we use the definition of logarithm to solve an exponential equation. a. 10 x = 5 x = log 5  Given equation Definition of logarithm this is the exact answer. Approximate calculator answer

25 Example 9 – Solution b. e –0.06x = –0.06x = ln  – Given equation Definition of logarithm Approximate calculator answer Exact answer; this can be simplified to in cont’d

26 Example 9 – Solution c. 8 x = x = log  Definition of logarithm; this is the exact answer Given equation Approximate calculator answer. Use the change of base theorem: cont’d

27 Example 9 – Solution Note: Many people will solve this by “taking the log of both sides”: 8 x = log 8 x = log x log 8 = log  Divide both sides by log 8. Take the “log of both the sides” Given equation This answer agrees with the first solution. cont’d

28 Exponential Equations Did you notice that the results of these two calculations in part c are the same? It simply involves several extra steps and some additional properties of logarithms. It is rather like solving quadratic equations by completing the square each time instead of using the quadratic formula.

29 Exponential Equations You can see that, before calculators, there were good reasons to avoid representations such as log Whenever you see an expression such as log , you know how to calculate it: log 156.8/log 8.