Copyright © Cengage Learning. All rights reserved. 4.2 LOGARITHMIC FUNCTIONS Copyright © Cengage Learning. All rights reserved. Introduction 3 Introduction In this section we introduce logarithmic functions, emphasizing the natural logarithm function. We then apply natural logarithms to a wide variety of problems, from doubling money under compound interest to drug dosage. 4 Logarithms 5 Logarithms The word logarithm (abbreviated log) means power or exponent. The number being raised to the power is called the base and is written as a subscript. For example, the expression log10 1000 means the exponent to which we raise 10 to get 1000. Since 103 = 1000, the exponent is 3, so the logarithm is 3: log10 1000 = 3 since 103 = 1000 We find logarithms by writing them as exponents and then finding the exponent. 6 Example 1 – FINDING A LOGARITHM Evaluate log10 100. Solution: Therefore log10100 = 2 since 102 = 100 7 Logarithms Logarithms to the base 10 are called common logarithms and are often written without the base, so that log100 means log10 100. We may find logarithms to other bases as well. Any positive number other than 1 may be used as a base. In general, for any positive base a other than 1, 8 Natural Logarithms 9 Natural Logarithms The most widely used of all bases is e, the number (approximately 2.718). Logarithms to the base e are called natural logarithms or Napierian logarithms. The natural logarithm of x is written ln x (“n” for “natural”) instead of loge x. In words: ln x is the power to which we raise e to get x. 10 Natural Logarithms The table below shows some values of the natural logarithm function f(x) = ln x, and its graph (based on these points) is shown on the right. 11 Natural Logarithms Notice that the graph of ln x is always increasing and has the y-axis as a vertical asymptote. The natural logarithm function may be used for modeling growth that continually slows. 12 Natural Logarithms The following graph shows logarithm functions for several different bases. Notice that each passes through the point (1, 0), since a0 = 1. We will concentrate on the natural logarithm function, since it is the one most used in applications. 13 Natural Logarithms Since logs are exponents, each property of exponents can be restated as a property of logarithms. The first three properties show that some natural logarithms can be found without using a calculator. 14 Example 4 – USING THE PROPERTIES OF NATURAL LOGARITHMS 15 Natural Logarithms The next four properties enable us to simplify logs of products, quotients, and powers. For positive numbers M and N and any number P: 16 Example 5 – USING THE PROPERTIES OF NATURAL LOGARITHMS 17 Doubling Under Compound Interest 18 Doubling Under Compound Interest How soon will money invested at compound interest double in value? The solution to this question makes important use of the property ln (MP) = P · ln M (“logs bring down exponents”). 19 Example 9 – FINDING DOUBLING TIME A sum is invested at 12% interest compounded quarterly. How soon will it double in value? Solution: We use the formula P(1 + r/m)mt with r = 0.12 and m = 4. Since double P dollars is 2P dollars, we want to solve 20 Example 9 – Solution cont’d The variable is in the exponent, so we take logarithms to bring it down. A sum at 12% compounded quarterly doubles in about 5.9 years. 21 Carbon 14 Dating 22 Carbon 14 Dating All living things absorb small amounts of radioactive carbon 14 from the atmosphere. When they die, the carbon 14 stops being absorbed and decays exponentially into ordinary carbon. Therefore, the proportion of carbon 14 still present in a fossil or other ancient remain can be used to estimate how old it is. 23 Carbon 14 Dating The proportion of the original carbon 14 that will be present after t years is 24 Example 12 – DATING BY CARBON 14 The Dead Sea Scrolls, discovered in a cave near the Dead Sea in what was then Jordan, are among the earliest documents of Western civilization. Estimate the age of the Dead Sea Scrolls if the animal skins on which some were written contain 78% of their original carbon 14. Solution: The proportion of carbon 14 remaining after t years is e–0.00012t. 25 Example 12 – Solution cont’d We equate this formula to the actual proportion (expressed as a decimal): e–0.00012t = 0.78 ln e–0.00012t = ln 0.78 –0.00012t = ln 0.78 Equating the proportion Taking natural logs ln e–0.00012t = –0.00012t by Property 3 Solving for t and using a calculator Therefore, the Dead Sea Scrolls are approximately 2070 years old. 26 Behavioral Science: Learning Theory 27 Behavioral Science: Learning Theory Your ability to do a task generally improves with practice. Frequently, your skill after t units of practice is given by a function of the form S(t) = c(1 – e–kt) where c and k are positive constants. 28 Example 13 – ESTIMATING LEARNING TIME After t weeks of training, your secretary can type S(t) = 100(1 – e–0.25t) words per minute. How many weeks will he take to reach 80 words per minute? Solution: We solve for t in the following equation: 100(1 – e–0.25t) = 80 Setting S(t) equal to 80 1 – e–0.25t = 0.80 Dividing by 100 –e–0.25t = –0.20 Subtracting 1 e–0.25t = 0.20 Multiplying by –1 29 Example 13 – Solution –0.25t = ln 0.20 cont’d Taking natural logs Solving for t Using a calculator He will reach 80 words per minute in about weeks. 30 Social Science: Diffusion of Information by Mass Media 31 Social Science: Diffusion of Information by Mass Media When a news bulletin is repeatedly broadcast over radio and television, the proportion of people who hear the bulletin within t hours is p(t) = 1 – e–kt for some constant k. 32 Example 14 – PREDICTING THE SPREAD OF INFORMATION A storm warning is broadcast, and the proportion of people who hear the bulletin within t hours of its first broadcast is p(t) = 1 – e–0.30t. When will 75% of the people have heard the bulletin? Solution: Equating the proportions gives 1 – e–0.30t = 0.75. Solving this equation gives t 4.6. Therefore, it takes about hear the news. hours for 75% of the people to 33