Section 2.3: Logarithmic Functions Lecture 3 MTH 124

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1 Procedural Skills Learning Objectives 1. Build an exponential function using the correct compounding identifiers (annually, monthly, continuously etc...) 2. Manipulate exponents algebraically. e.g. Solving problems of the form x 50 = Manipulate logarithms algebraically. e.g. Solving problems of the form 2 t = 15 or e x = Translate a rate given as percentage to its decimal rate. e.g. 3% continuous exponential decay give us r = 0.03 Interpretation Skills 1. Understand the relationship between logarithmic and exponential functions. 2. Recognize clues that indicate when an exponent or logarithm rule is appropriate to solve a problem. 3. Solve a variety of procedural and contextual questions related to logarithms/exponents. 4. Compare and contrast doubling time and half-life. What do all half-life problems have in common? What do all doubling time problems have in common? 5. Construct an exponential function from a context. It is natural when learning about a particular function to wonder about the function s inverse. Recall that intuitively the inverse of a function undoes what the function did. In other words, if a function f(x) sends some value x to another value y then the inverse of f will send y back to x. Put another, way if f(x) = y then f 1 (y) = x. Note that the superscript notation, f 1 (x), means the inverse of f(x). The inverse of the exponential function is given by the logarithm. log b x The base b logarithm of x, denoted log b x, is the power that we need to raise b to get x. Example 1 To understand the logarithm lets consider the toy problem log 2 (9). This notation log 2 (9), read log of 9 base 2, is asking the following question. 2 to what power will give us 9? This can be written as 2? = 9. Fortunately we can use our calculators to calculate logarithms. So to calculate log 2 (9) on a TI-83 we would input this as log(9)/ log(2). This comes from logarithm properties. Please make sure you review these properties on page 157 of the text. To make things more straight-forward we will generally use log base 10 or the natural log, denoted ln. log base 10 is the default for your calculator. So when you see log(3) this is equivalent to log 10 (3) and can be calculated by pressing LOG, 3, and enter. a a We re interested in understanding these properties because the exponential and logarithm functions occur surprisingly often when modeling the phenomenon from the natural world. 14

2 Natural Log In section 2.2 we used the exponential equation P e rt to describe continuous growth. This type of growth is often used to model the growth of bacteria, population, epidemics, and countless other phenomena. 1 Example 2 Suppose the model A(t) = 1.514e 0.05t describes how many thousand people are infected with Ebola in Sierra Leone t months after August As a public health official we may be interested to know when we expect the number of infected people to reach a certain amount. For example, how long will it take for 10,000 people to be infected? Doubling Time & Half-Life When we have exponential growth a natural quantity of interest is the time it will take for our original quantity to double. Example 3 Suppose we invest $ 3,000 dollars into an account which pays an annual interest rate of 3% per year. When will our money be doubled? 1 It is more intuitive to model exponential biological growth using continuous compounding. For example, a growing colony of bacteria doesn t just double in size in an instant. 15

3 ln(x) The natural logarithm of some number x, written ln x, is the power of e needed to get x. aa Note that ln(1) = 0 and ln(e) = 1, as you will use these facts often. In the case of exponential decay we re often interested in the time it takes for our original amount to half. Fortunately the idea is the same as doubling time. Example 4 Suppose we have invested in a stock whose value seems to be decaying exponentially. The stock peaked at $50 per share and since has been decaying according to the formula V (t) = 50e 0.01t where V (t) is dollars per share t days after the peak value. We ve decided that if this trend continues we ll sell the stock when it reaches half its peak value. According to our model when will the peak stock value be halved? Summary 1. What is the meaning of the notation log(x), ln(x)? What are important related formulas? 2. What kind of problems do we use these tools to solve? 3. What is the relationship between ln and e? What are two important formulas that illustrate their relationship? 4. Given exponential growth, doubling time is given by Given exponential decay, half-life is given by... 16

4 Carbon Dating Example The half-life of the radioactive isotope Carbon-14, denoted 14 C, is 5,200 years. The concentration of Carbon-14 in a living organic body is equivalent to that of the atmosphere. Since we know atmospheric 14 C concentrations have been relatively constant, scientists can use this knowledge to determine the age of an organic material Suppose an organism begins with 50 g of 14 C. Construct a function, A(t), which gives the amount of P g of 14 C, t years after the organism has died. Note: Since all 14 C decays at the same rate you may use the base/rate you find here for the remainder of the problems. 2. Approximately how many grams of 14 C did an organism initially possess if there are 7 g remaining after 17,830 years? 2 This is one of many dating methods that adds to the immense body of evidence scientists have gathered about the age of plants, animals, the earth, and the universe. The earth is approximately billion years old! 17

5 For problems 3-4 no initial value is provided or necessary to solve the problem. 3. If 12% of the initial amount of 14 C in a sample remains, how much time has elapsed? 4. Approximately what percentage of an initial sample of 14 C remains after 33,450 years? Note that these problems all have to do with continuous exponential decay in one particular context. Make sure you try a variety of different problems involving exponential and logarithmic functions in different contexts! 18