VIEWS: 0 PAGES: 16 POSTED ON: 2/11/2013 Public Domain
L5.5 Logarithmic Functions a logarithm is an exponent a logarithm is an exponent a logarithm is an exponent a logarithm is an exponent L5.5 Warm up Find the value of x • 10x = 100,000 x=5 • 2x = 1/64 x = –6 • ex = e1/x x = 1/x → x2 = 1; x = ±1 • 4x = –1 no soln • 4x = 2048 22x = 211 2x = 11 x = 11/2 When you can equate bases, exponents can be equated. If you can not equate bases, you need logarithms… Logarithmic Functions f(x) = 3x • Sketch graphs of f(x) = 3x and f-1(x) f-1(x) = log3 x x f(x) x f-1(x) -2 1/9 1/9 -2 -1 1/3 (0, 1) 1/3 -1 0 1 1 0 1 3 (1, 0) 3 1 2 9 9 2 The inverse of the exponential function y = bx is x = by and x = by is the logarithmic function y = logb x. x = by ↔ y = logb x Logarithmic Functions f(x) = bx f-1(x) = logb x Notice that the bases, b, are (0, 1) the same for the inverses and that b > 0, b ≠ 1. (1, 0) Exponential functions (with a = 1): Logarithmic functions: Are continuous and one-to-one Are continuous and one-to-one Domain: {x | x is Real} Domain: {x | x > 0} Range: {y | y > 0 } Range: {y | y is Real } Contain the point (0, 1) Contain the point (1, 0) y = 0 is a horizontal asymptote x = 0 is a vertical asymptote Because exponential and logarithmic functions are inverses of each other, they swap their domains and ranges. The restriction on the domain of logs is very important. Why can’t it be negative? Try log(-5) or ln(-5) in your calculator. What happens? Logs are just Exponents! x = by ↔ y = logb x Base is 4. Examples: 64 = 43 ↔ 3 = log4 64 Exponent is 3. Result is 64. Base is 2. 32 = 25 ↔ 5 = log2 32 Exponent is 5. Result is 32. Base is 10. 100 = 102 ↔ 2 = log10 100 Exponent is 2. Result is 100. You should be able to convert between exponential and logarithmic forms: log2 16 = 4 24 = 16 log10 1000 = 3 103 = 1000 62 = 36 log6 36 = 2 34 = 81 log3 81 = 4 Finding Logs Two approaches: 1) If the input to the log function is a power of the base, you can find the log without a calculator. 2) If the input to the log function is not a power of the base, you need a calculator to find the log. Finding Logs (when input is a power of the base) These can be done without a calculator. Solve: “What exponent do I raise 10 to to get 10,000?” x = log10 10,000 x = 4 because 104 = 10,000 “What exponent do I raise 5 to to get 1/125?” x = log5 1/125 x = -3 because 5-3 = 1/125 Do these: Note: log6 63 = 3 and 10log10 4 = 4 1. x = log6 216 x=3 same bases 2. x = log7 1/49 x = -2 logb bx = x and blog b x = x 3. x = log10 0.0001 x = -4 Logs & exponentials are inverses! 4. x = log4 –4 No solution. Domain > 0! (There is no exponent that I can raise 4 to to get -4) Finding Logs (when input is not a power of the base) These require a calculator. Calculators have logs for 2 bases built in: 10 and e*. log() on Ti89 log10 is the Common Log. log x without a base means log10 x Is “hotkey” ◊7 log 75 means “what do I raise 10 to to get 75”? log 75 ≈ 1.875 b/c 101.875 ≈ 75 It’s between 1 & 2. Loge is the Natural Log. ln x means loge x. Why? ln 32 means “what do I raise e to to get 32”? ln 32 ≈ 3.466 b/c e3.466 ≈ 32 Do these (to 2 decimal places): You should check 1. x = log 6 x = 0.78 your answers! 2. x = log 0.03 x = –1.52 * We will learn how to use the 3. x = ln 104 x = 4.64 calculator for bases other than 4. x = ln 0.8 x = – .22 10 and e later. Homework (due Tuesday) pp 194-195 #1 – 9 odd, 10, 11 – 17 odd, 28, 29, 34 (mostly odds so you can check your homework). You are encouraged to read Lesson 5.5 if you are confused about logs and/or you wait to do your homework until Monday night. Remember: a logarithm is an exponent a logarithm is an exponent a logarithm is an exponent a logarithm is an exponent L5.5 Checkpoint after day 1 • A logarithm is just ________________ • Find f-1(x) if f(x) = 7x • What is the domain of f-1(x)? Why? What is the range? Why? • Convert between log and exp’l forms: 1. log 500 = x 2. ex = e5 3. ex = 200 4. log 1000 = x • Solve the above problems (which ones need a calculator?) L5.5 Summary (so far): • A logarithms is just an exponent • The inverse of f(x) = bx is f-1(x) = logb x • The domain of a log function is {x | x > 0} Why? • You can convert between logarithmic and exponential forms x = by ↔ y = logb x • To find a logarithm: • If the input is a power of the base, just return the exponent • If the input is not a power of the base, use your calculator • common log: log x means log10 x • natural log: ln x means loge x • we’ll learn later how to find logs of other bases • Next: Applications of logs • Logs are used when the range of information is huge Wrap Up (day 1) or Warm Up (day 2) ≈8 0.9031 10 Class exercises p194 #1-9 #3. Given log 8 ≈ 0.9031. [How could you check this?] Find log 80 and log 0.08 without a calculator. log 8 = 0.9031 → 100.9031 = 8 10???? = 80 ← log 80 10*100.9031 = 8*10 101.9031 = 80 log 80 = 1.9031 log 8 = 0.9031 → 100.9031 = 8 10???? = 0.08 ← log 0.08 10-2*100.9031 = 8*10-2 10-1.0969 = 0.08 log 0.08 = -1.0969 Decibels: An Application of Logs Decibels (dB) is a measure of the loudness of sound. The decibel scale is an example of a logarithmic scale*. Given a sound intensity level, I, I dB = 10 log , where I0 is the intensity of a barely audible sound. I0 [Page 192 has a table with intensity levels for common sounds.] * There are many others: Examples: brightness, acidity, & 1. Average car @ 70kmh, I = 106.8I0. dB? earthquake magnitude. I 106.8 I 0 10 log 10 log (10) log106.8 68dB I0 I0 I person whispering: 15dB 2 people 18dB 2. Whisper, I = 101.5I0. dB of 2 people whispering? I 2 101.5 I 0 10 log 10 log (10) log(2 101.5 ) 18dB I0 I0 You do not have to memorize these defintions. Richter Scale: An Application of Logs • Earthquakes are measured on the Richter scale which is a logarithmic scale. • The Magnitude, M, of an earthquake on the Richter scale is as follows: 2 E M log 3 E0 where E is the energy released by the earthquake measured in joules and E0 = 104.4 joules is the energy released by a very small reference earthquake. 1. The 1906 San Francisco earthquake released approximately 5.96 X 1016 joules of energy. What was the magnitude of this earthquake on the Richter scale? 2 E 2 5.96 X 1016 M log 4.4 log 4.4 8.25 3 10 3 10 Richter Scale: An Application of Logs 2. Generally, an earthquake requires a magnitude over 5.6 on the Richter scale to inflict serious damage. How many times more powerful than this was the great 1906 Columbia earthquake, which registered a magnitude of 8.6 on the Richter scale? 1906 Columbia Earthquake Earthquake with magnitude 5.6 2 E 2 E log 8.6 log 5.6 3 E0 3 E0 E E log 4.4 12.9 log 8.4 4.4 10 10 E E 10 12.9 108.4 10 4.4 104.4 E 1017.3 joules E 1012.8 joules 1017.3 12.8 32,000 times more powerful 10 Logs are Exponents! If a log is an expression with multiple terms, you can use the rules of exponents to simplify Ex: If ln y = 4x + 2 show that y ≈ 7.4(54.6)x y = e4x+2 convert to exponential form y = e4x · e2 use rules of exponents y = e2 · (e4)x use rules of exponents y ≈ 7.4(54.6)x estimate e2 and e4