10 THE NATURE OF GROWTH Copyright © Cengage Learning. All rights reserved. 10.1 Exponential Equations Copyright © Cengage Learning. All rights reserved. Exponential Equations 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. 3 Exponential Equations In this section, we investigate solving equations known as exponential equations An exponential is an expression of the form bx; The number b is called the base. In definition bx is defined as an integer. 4 Exponential Equations Let’s solve the exponential equation 2x = 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: x = 2: x = 3: x = 4: 2x = 21 = 2 2x = 22 = 4 2x = 23 = 8 2x = 24 = 16 Too small Too small Still too small 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. 5 Definition of Logarithm 6 Definition of Logarithm The solution of the equation 2x = 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 7 Definition of Logarithm We further shorten this notation to x = exp214 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. 8 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 = exp214 and x = log214 mean exactly the same thing. 9 Definition of Logarithm This leads us to the following definition of logarithm. The statement x = logb 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. 10 Example 2 – Write exponentials in logarithmic form Write in logarithmic form: a. 52 = 25 Solution: a. In 52 = 25, 5 is the base and 2 is the exponent, so we write 2 = log5 25 Remember, the logarithmic expression “solves” for the exponent. 11 Example 2 – Solution b. With cont’d = 2–3, the base is 2 and the exponent is –3: –3 = log2 c. With (since = 8, the base is 64 and the exponent is = 641/2 ): = log64 8 12 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.” 13 Definition of Logarithm The solution for the equation 10x = 2 is x = log 2, and the solution for the equation ex = 0.56 is x = ln 0.56. 14 Evaluating Logarithms 15 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 = logb A means bx = 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. 16 Example 6 – Evaluate logarithmic expressions Evaluate the given expression. Solution: a. From the definition of logarithm, log2 8 = x means 2x = 8 or x = 3. Thus, log2 8 = 3. By calculator, Also, 17 Evaluating Logarithms 18 Example 7 – Evaluate logarithms with a change of base Evaluate (round to the nearest hundredth): a. log7 3 b. log3 3.84 Solution: 19 Evaluating Logarithms We now return to the problem of solving 2x = 14 Given equation. x = log2 14 Solution We call log2 14 the exact solution for the equation, and Example 8 finds an approximate solution. 20 Example 8 – Solve an exponential equation Solve 2x = 14 (correct to the nearest hundredth). Solution: We use the definition of logarithm and the change of base theorem to write 21 Exponential Equations 22 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. 23 Example 9 – Solve exponential equations with common and natural logs Solve the following exponential equations: a. 10x = 5 b. e–0.06x = 3.456 c. 8x = 156.8 Solution: Regardless of the base, we use the definition of logarithm to solve an exponential equation. a. 10x = 5 x = log 5 0.6989700043 Given equation Definition of logarithm this is the exact answer. Approximate calculator answer 24 Example 9 – Solution b. e–0.06x = 3.456 –0.06x = ln 3.456 cont’d Given equation Definition of logarithm Exact answer; this can be simplified to in 3.456. –20.66853085 Approximate calculator answer 25 Example 9 – Solution c. 8x = 156.8 x = log8 156.8 2.43092725 cont’d Given equation Definition of logarithm; this is the exact answer Approximate calculator answer. Use the change of base theorem: 26 Example 9 – Solution cont’d Note: Many people will solve this by “taking the log of both sides”: 8x = 156.8 log 8x = log 156.8 Given equation Take the “log of both the sides” x log 8 = log 156.8 Divide both sides by log 8. 2.43092725 This answer agrees with the first solution. 27 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. 28 Exponential Equations You can see that, before calculators, there were good reasons to avoid representations such as log8 156.8. Whenever you see an expression such as log8 156.8, you know how to calculate it: log 156.8/log 8. 29