What on Earth is a Logarithm? A logarithm is just an exponent. To be specific, the logarithm of a number x to a base b is just the exponent you put onto b to make the result equal x. For instance, since 5² = 25, we know that 2 (the power) is the logarithm of 25 to base 5. Symbolically, log5(25) = 2. Where Did Logs Come From? Before pocket calculators, you needed logs to compute most powers and roots with fair accuracy. Most math books contained log tables to look up values. The invention of logs in the early 1600s fueled the scientific revolution. Back then scientists, astronomers especially, used to spend huge amounts of time crunching numbers on paper. The slide rule, once almost a cartoon trademark of a scientist, was nothing more than a device built for doing various computations quickly, using logarithms. By using logarithms, exponential problems become multiplication problems, multiplication problems become addition problems, and division problems become subtraction problems. Why Do We Care? Why Do We Use Logarithms? To find the number of payments on a loan or the time to reach an investment goal. To model many natural processes, particularly in living systems. We perceive loudness of sound as the logarithm of the actual sound intensity, and dB (decibels) are a logarithmic scale. We also perceive brightness of light as the logarithm of the actual light energy, and star magnitudes are measured on a logarithmic scale. To measure the pH or acidity of a chemical solution. To measure earthquake intensity on the Richter scale. To analyze exponential processes. The log function is the inverse of the exponential function. Applications include cooling of a dead body, growth of bacteria, and decay of a radioactive isotopes. The spread of an epidemic in a population often follows a modified logarithmic curve called a “logistic”. Why base e? What’s so special about e? Most of the explanations need some calculus; however, e has a definition in terms of factorials: e = 1/0! + 1/1! + 1/2! + 1/3! + ... Numerically, e is about 2.7182818284. It’s irrational (the decimal expansion never ends and never repeats), and in fact like pi it’s transcendental (no polynomial equation with integer coefficients has pi or e as a root.) Example 1) Population Increase (exponential growth) The world population (in millions) from 1992 through 2000 is shown in the table. Year 1992 1993 1994 1995 1996 1997 1998 1999 2000 Pop 5450 5531 5611 5691 5769 5847 5925 6003 6080 An exponential growth model that approximates these data is given by P 5000 10,303.893e0.00730t where P is the population (in millions) and t=2 represents 1992. According to the model, when will the world population reach 7.1 billion? In the year 2012! Example 2) Finding an exponential growth model Find an exponential growth model whose graph passes through the points (0,4453) and (7,5024). The general form of the model is y aebx . y 4453e 0.01724 x Example 3) Carbon Dating (exponential decay) In living organic material, the ratio of radioactive carbon isotopes (carbon 14) to the number of nonradioactive carbon isotopes (carbon 12) is about 1 to 10^12. When organic material dies, its carbon 12 content remains fixed, whereas the carbon 14 begins to decay with a half-life of about 5700 years. To estimate the age of dead organic material, scientists use the following formula to denote this ratio: 1 t 8223 R 12 e 10 1 The ratio of carbon 14 to carbon 12 in a newly discovered fossil is R 13 . Estimate the age of the fossil. 10 The fossil is about 18,934 years old! Example 4) SAT scores (Gaussian model) In 2001, the SAT scores for males in the United States roughly followed a normal distribution (bell-curve) given by ( x 533) 2 y 0.0035e 26,450 where x is the SAT score for mathematics. Approximate the distribution of students who score 650 on the mathematics test. The distribution is 0.0020859418! Example 5) Spread of a virus (logistics growth model) A logistics curve shows rapid population growth followed by a declining rate of growth. On a college campus of 5000 students, one student returned from vacation with a contagious flu virus. The spread of the virus through the student population is given by 5000 y 1 4999e0.8t where y is the total number infected after t days. The college will cancel classes when 40% or more of the students are ill. a) How many are infected after 5 days? 54 students b) After how many days will the college cancel classes? 10 days Example 6) Magnitudes of Earthquakes (logarithmic model) On the Richter scale, the magnitude R of an earthquake of intensity I is given by I R log10 I0 where Io=1 is the minimum intensity used for comparison. Find the intensity per unit of area for the following earthquakes: a) Prince William Sound, Alaska in 1964, R = 9.2 1,584,893,000 b) Near Kodiak Island, Alaska in 2001, R = 7.1 12,589,000 How does the intensity of the earthquakes compare? The Prince William Sound earthquake was 126 times greater than that of Kodiak Island.
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