5.2 Exponential and Logarithmic Functions in Finance

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1 5. Exponential and Logarithmic Functions in Finance Question 1: How do you convert between the exponential and logarithmic forms of an equation? Question : How do you evaluate a logarithm? Question 3: How do you solve problems using logarithms? Many applications in business are based upon a rate of growth or decay given as a percentage. For instance, the rate at which the value of a $150,000 home is increasing may be given as % per year. The value of the home t years later would be Vt ( ) 150, t Other quantities will decrease by a percentage. At the completion of a marketing campaign, sales for a new tablet computer are 550,000. Sales decrease by 5 percent per week from that point onward. The sales t weeks later may be modeled by St ( ) 550, t Each of these quantities are modeled by an exponential function of the form 1 f t a r where a is the original amount of the quantity and r is the rate in percent per time. The time should correspond to the units of time on the variable t. The plus sign is used when the function is an increasing exponential function. The minus sign is used when the quantity is decreasing. In this section, you ll learn how to use logarithms to solve equations arising from exponential functions. t 1

2 Question 1: How do you convert between the exponential and logarithmic forms of an equation? Exponential and logarithm functions are inverses of each other. In the simple terms, this means that an expression in exponential form may be converted to logarithm form by switching the inputs and outputs. Let s start with a concrete example. The exponential function y 10 x takes the variable x as its input and outputs the variable y. For an input of x we get an output of y 100 since On a logarithm of base 10 (called a common logarithm), these roles are reversed. The common logarithm must take in y 100 and output x, 10 log 100 For common logarithms, those with base 10, the base on the logarithm is often left out and written as log 100 This means that whenever you see a logarithm without a base, it is assumed to have a base of 10. Let s compare these forms side by side. Exponential Form Logarithmic Form log 100 input output output input 10

3 The base on the exponential form is below the. On the logarithm form, the base is just after and slightly below the word log. The exponential form take in and outputs 100. The logarithm form does exactly the opposite. It takes in 100 and outputs. The numbers are the same, but they role they play is reversed. For all bases b 0, y x b means that x y log b Example 1 Convert to Logarithmic Form Convert each exponential form below to its equivalent logarithmic form. a Solution For this exponential form, the base is 10 so it will convert to a logarithm base 10 or common logarithm. The input is 3 and the output is When this exponential is converted to a logarithm, the input will be 1000 and the output will be 3. This gives the logarithmic form, 3 log 1000 Note that the original exponential form has the exponent on the left side instead of the right. Where it appears is irrelevant. They key is to recognize that the exponent is the input and the other side of the form is the output. b. 5 3 Solution This exponential form will convert to a logarithm with a base of. Since the input is 5 and the output is 3, the logarithmic form is log 3 5 3

4 c. 5 t z Solution Even though the input and output have variables, we may still reverse the roles and write logarithmic form as log5 z t The base e occurs frequently in business and finance. Like the constant π, the constant e represents an irrational number whose value is approximately.718. Because this is an approximation, you ll often see the approximately equal sign,, when using this base. Most scientific and graphing calculators have an e button to help you evaluate exponentials with a base of e. Exponential forms with a base of e convert to logarithms with a base of e. For instance, Exponential Form Logarithmic Form e log e output input output input Logarithms with a base of e are also called natural logarithms. They are often abbreviated by writing ln instead of log e Example Convert to Logarithmic Form Convert each exponential form with a base of e to logarithmic form. a. 0 e 1 Solution The logarithmic form for this exponential form is 4

5 ln 1 0 b. A e rt P Solution In this exponential form, groups of variables play the role of input and output. The input in the exponential form is rt and the output is the fraction A P. The corresponding natural logarithm is rt ln A P Logarithmic forms may also be converted to exponential form. As with exponential form, they key is to identify the input and output and switch those roles. Example 3 Convert to Exponential Form Convert each logarithmic form to exponential form. a. log Solution This form converts to an expopnential form with a base of 4. The input on the logarithmic form is 64 and the output is 3. The corresponding exponential form is b. y b ln x Solution Before identifying the input and output, isolate the logarithm. This is done by subtracting b from both sides to yield yb ln x 5

6 In this form, the input to the logarithm is x and the output is y b. The exponential form is e yb x c. M log A A0 Solution The input on this common logarithm is the group output is M. Switching these roles gives the exponential form A A 0 and the A 10 M A 0 6

7 Question : How do you evaluate a logarithm? Many logarithms may be calculated by converting them to exponential form. Suppose we want to calculate the value of log logarithmic form, 16. Start by writing this expression as a log 16? We could write the output with a variable, but a question mark suffices to indicate what we want to find. If we convert this form to an exponential form with a base of,? 16 The left hand side may be written with the base as place of 16,? 4 4. Substitute this expression in Since the exponent on the left side must be 4, this is also the value in the original exponential form, log 16 4 This strategy works well as long as we can write the number on the right with the same base as the exponential on the other side of the equation. Example 4 Evaluate the Logarithm Find the value of each logarithm by converting to exponential form. a. log9 81 Solution Write the logarithm in logarithmic form and convert to exponential form, 9? log 81? means that

8 Since 9 81, the value of the logarithm is, log b. log3 7 Solution The logarothmic and exponential forms are 1? 1 log 3? means that Since , the value of the logarithm is -3, log3 3. The 7 7 negative power makes the reciprocal. c. ln e Solution The logarothmic and exponential forms are ln e? means that e e? The value of the logarithm is, ln e. In effect, the number is put into the base of e and the natural logarithm reverses this process. Not every logarithm may be solved by converting to exponential form. For this strategy to work, we must be able to write each side of the exponential form with the same base. Scientific and graphing calculators are both able to calculate natural logarithms and common logarithms. Natural logarithms are calculated using a button labeled something like LN. Using this button, you should be able to do the following calculations by pressing the LN button, entering the number, and pressing the ENTER or = button. 8

9 ln 1 0 ln ln You may also calculate common logarithms in a similar manner using a button that is typically labeled LOG. Using this button, you should be able to compute each of the following common logarithms. log 10 1 log log Using these buttons, you can compute any natural or common log. Even the logs that may not be solved by converting to exponential form may be computed on a calculator. Some calculator may even have a button for calculating a logarithm with any positive base. To see if your calculator is able to do this, consult the manual for your calculator. If your calculator does not have this button, you can use the change of base formula to compute logarithms with any positive base. Change of Base Formula for Logarithms For any positive base a and b not equal to 1, log a x log log b b x a where x is a positive number. This formula is used to compute a logarithm with base a by converting it to two logarithms with base b. The base b can be any positive number not equal to 1, but usually it is a base of 10 or e so that a calculator may be used to compute the right hand side of the formula. 9

10 Example 5 Compute the Logarithm Find the value of each logarithm using the Change of Base formula for Logarithms. a. log7 10 Solution We may use the Change of Base formula to convert this logarithm to natural logarithms or common logarithms. If we convert to natural logarithms we get 7 log 10 ln 10 ln A calculator is used to evaluate the natural logarithms. The values of the individual logarithms are shown above, but it is a good idea to type the entire expression. This avoids rounding in the middle of the problem and then rounding again at the end. Ideally you should only round once. If we convert to common logarithms, 7 log 10 log 10 log The value of the original base 7 logarithm is the same whether it is computed from natural logs or common logs. b. log

11 Solution Use the Change of Base formula with natural logarithms to give 1.05 log 100 ln 100 ln You may also calculate the value using common logarithms, 1.05 log 100 log 100 log

12 Question 3: How do you solve problems involving using logarithms? Logarithms are useful for solving exponential equations. An example of an exponential equation is the equation x To solve this equation for the variable x, isolate the term containing the exponential piece. This is done by subtracting from both sides of the equation to give x We remove the variable from the exponent by converting this exponential form to logarithmic form. The logarithmic form is Divide both sides of the equation by to yield x log 14 log 14 x This is the exact solution to the original exponential equation. We can use this expression to find an approximate solution to as many decimal places as needed. To three decimal, the solution is x If we had worked out the logarithm earlier in the calculation, we would not be able to write down the exact solution. The best practice is to find the exact solution first. Use the exact solution to get an approximate solution. Example 6 Compound Interest How long will it take$10,000 to double in an account earning % compounded quarterly? Solution For this problem, we ll use the Compound Interest Formula FV PV 1 i n 1

13 Since we want to know how long it will take, let t represent the time in years. The number of compounding periods is four times the time or n 4t. The original amount is PV 10,000 and the future value is double or FV 0, 000. The interest rate per period is 0.0 i When these values are substituted into the compound interest formula, we get the exponential equation t To solve this equation for t, isolate the exponential factor by dividing both sides by 10,000 to give t Convert this exponential form to logarithm form and divide by 4, t log log1.005 t 4 To find an approximate value, use the Change of Base Formula to convert to a natural logarithm (or a common logarithm): t ln ln years It is interesting to note that the starting amount is irrelevent when doubling. If we started with P dollars and wanted to accumulate P at the same interests rate and compounding periods, we would need to solve 13

14 4 P P t This reduces to the same equation as above, t when both sides are divided by P. This means it takes about 37.4 years to double any amount of money at an interest rate of % compounded quarterly. In this example, converting to logarithm form removes the variable from the power in the exponential factor. This makes it easy to solve for the variable. This same strategy is used to solve for other variables in the power of an exponential also. Example 7 Continuous Exponential Decay The value of a large piece of equipment depreciates from $15,000 to $50,000 in five years. If the value decreases exponentially, at what continuous rate is the value dropping? Solution We will model the value V of the equipment at some time t years later with the equation V V0 e rt In this equation, the original value of the equipment is V 0. The value is decreasing at a continuous rate of r (as a percent) due to the negative sign in the power. Put the values in this equation to yield r 5 e Solve for the rate r by converting to logarithmic form: 14

15 e 5r r ln ln r 5 Divide both sides by to isolate the exponential factor Convert to logarithmic form Divide both sides by 5 This is the exact rate which may be evaluated and rounded to three decimal places to give r The equipment is depreciating at a continuous rate of approximately 18.3 percent per year. Note that the rate is always given as a percent per some time period. This time period is the same as the units on the variable t. 15