§3.3 Logarithmic Functions. The student will learn about: the logarithmic function, its properties, and applications.
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 carbon 14 dating.
The Logarithmic Function Definition. The inverse of an exponential function is called a logarithmic function. That is, where b > 0, and b ≠ 1; if y = log b x , then b y = x . The domain of the log function is the set of all positive real numbers and the range of the log function is the set of all positive real numbers.
Examples If y = log b x , then b y = x Log 3 9 = 2 Since 3 2 = 9 Log 2 16 = y, find y. So, y = 4. Log 5 x = 3, find x. So, x = 125. Log b 1/8 = -3, find b. So, b = 2.
Calculator Evaluation of Logarithms Common logarithms: log x = log 10 x. Note that log x = y is equivalent to x = 10 y and the inverse is y = 10 x. log 2 = 0.30103 log 12.345 = 1.09149 log 0.2001 = - 0.69875 Graph y = logx. 0 x 10 -5 y 3. Try any of the above on your calculator if you wish.
Calculator Evaluation of Logarithms Natural logarithms: ln x = log e x. Note that ln x = y is equivalent to x = e y and the inverse is y = e x. ln 2 = 0.69315 ln 12.345 = 2.51325 ln 0.2001 = - 1.60894 Graph y = ln x. 0 x 10 -5 y 2. Try any of the above on your calculator if you wish.
Properties Of A Logarithmic Graph. Unless shifted. From the graph y = logx. • Pass through (1, 0). • The graph is continuous. • The graph is asymptotic to the y axis. • The graph passes through ( ... , (b –2, -2), (b –1, -1), (b, 1), (b 2, 2), (b 3, 3), …) etc. • If b > 1 (almost always true) the graph is increasing. • If 0 < b < 1 the graph is decreasing.
Properties of Logarithmic Functions. • log b 1 = 0. • log bb = 1. • log b b x = x. This is somewhat useful. These four properties follow directly from the definition of logarithm.
Properties of Logarithmic Functions. • log b MN = log b M + log b N. • log b M/N = log b M - log b N. • log b MP = P · log b M. These four properties are useful in solving logarithmic equations.
Useful Information This means that
Application - Doubling Your Money After graduating from York College, Sam Spartan landed a great job with Springettsbury Manufacturing, Inc. His first year he bought a $5,000 Roth IRA and invested it in a stock sensitive mutual fund that grows at 10% a year. How long will it take for that investment to double? A = P e rt OR 10,000 = 5000 e 0.10t AND solve for t. 10000/5000 = e 0.10t or 2 = e 0.10t But 0.10 t = ln 2 so t = ln 2/ .10 = 6.93 years Remind the students of the “Rule of 72”.
Application 2 Page 121 #103. Using the formula, N = 10 log (I/I 0), and the fact that I0 = 10 –16 watts/cm2, find the decibel ratings of the following sounds: a. Whisper: 10 –13 watts/cm2. N = 10 log (I/I 0) = 10 log 10 3 watts/cm2 = 30
Application 2 (continued) Page 121 #103. Using the formula, N = 10 log (I/I 0), and the fact that I0 = 10 –16 watts/cm2, find the decibel ratings of the following sounds: a. Normal conversation: 3.16 · 10 –10 watts/cm2. = 10 log 3.16 · 10 6 watts/cm2 = 65 NOTE: There are many quantities in real life that are related in a logarithmic manner.
Leaves rustling Ears ringing all over campus How loud is too loud? Hearing loss occurs after eight hours of exposure to noises over 85 decibels. The length of exposure is cut in half with each additional three decibels. Any sound above 130 causes immediate harm. Car horn Jet engine up close Conversation Whisper 10 30 54 63 80 96 120 160 Library reading room Traffic at George and Country Club Wiz Khalifa in concert.
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.
Carbon 14 Dating The proportion of the original carbon 14 that will be present after t years is Half-life
Application 3 – 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 and is 78%.
cont’d Application 3 – Solution We equate this formula to the actual proportion (expressed as a decimal): e–0.00012t = 0.78 Take the ln of each side. ln e–0.00012t = ln 0.78 Simplify. – 0.00012t = ln 0.78 Divide by – 0.00012. ≐ 2071 And solve.. Therefore, the Dead Sea Scrolls are approximately 2070 years old.
Summary. • We learned about the logarithmic function. • We learned common logs base 10 and natural logs base e. • We learned about financial problems that involve the logarithmic function. • We learned about other problems that involve the logarithmic function.
ASSIGNMENT §3.4 on my website. 7, 8, 9, 10, 11, 12, 13, 14, 15