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L5.5 Logarithmic Functions

L5.5 Logarithmic Functions. a logarithm is an exponent. a logarithm is an exponent. a logarithm is an exponent. a logarithm is an exponent. Find the value of x 10 x = 100,000 2 x = 1/64 e x = e 1/x 4 x = –1 4 x = 2048. L5.5 Warm up. x = 5. x = – 6.

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L5.5 Logarithmic Functions

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  1. L5.5 Logarithmic Functions a logarithm is an exponent a logarithm is an exponent a logarithm is an exponent a logarithm is an exponent

  2. Find the value of x 10x = 100,000 2x = 1/64 ex = e1/x 4x = –1 4x = 2048 L5.5 Warm up x = 5 x = –6 x = 1/x → x2 = 1; x = ±1 no soln 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…

  3. Logarithmic Functions f(x) = 3x f-1(x) = log3 x • Sketch graphs of f(x) = 3x and f-1(x) (0, 1) (1, 0) 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

  4. Because exponential and logarithmic functions are inverses of each other, they swap their domains and ranges. Logarithmic Functions f-1(x) = logb x f(x) = bx (0, 1) (1, 0) Notice that the bases, b, are the same for the inverses and that b > 0, b ≠ 1. Exponential functions (with a = 1): Are continuous and one-to-one Domain: {x | x is Real} Range: {y | y > 0 } Contain the point (0, 1) y = 0 is a horizontal asymptote Logarithmic functions: Are continuous and one-to-one Domain: {x | x > 0} Range: {y | y is Real } Contain the point (1, 0) x = 0 is a vertical asymptote 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?

  5. Logs are just Exponents! x = by↔ y = logb x Base is 4. Exponent is 3. Result is 64. You should be able to convert between exponential and logarithmic forms: Examples: 64 = 43↔ 3 = log4 64 Base is 2. Exponent is 5. Result is 32. 32 = 25↔ 5 = log2 32 Base is 10. Exponent is 2. Result is 100. 100 = 102↔ 2 = log10 100 log2 16 = 4 log10 1000 = 3 62 = 36 34 = 81 24 = 16 103 = 1000 log6 36 = 2 log3 81 = 4

  6. Finding Logs Two approaches: If the input to the log function is a power of the base, you can find the log without a calculator. If the input to the log function is not a power of the base, you need a calculator to find the log.

  7. Finding Logs (when input is a power of the base) These can be done without a calculator. Solve: Note: log6 63 = 3 and 10log 4 = 4 10 same bases logb bx= x and blog x = x b “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: • x = log6 216 • 2. x = log7 1/49 • 3. x = log10 0.0001 • 4. x = log4–4 x = 3 x = -2 x = -4 No solution. Domain > 0!(There is no exponent that I can raise 4 to to get -4) Logs & exponentials are inverses!

  8. 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*. log10 is the Common Log. log x without a base means log10 x log 75 means “what do I raise 10 to to get 75”? log 75 ≈ 1.875 b/c 101.875 ≈ 75 Loge is the Natural Log. ln x means loge x. ln 32 means “what do I raise e to to get 32”? ln 32 ≈ 3.466 b/c e3.466 ≈ 32 log() on Ti89 Is “hotkey” ◊7 It’s between 1 & 2.Why? x = 0.78 x = –1.52 x = 4.64 x = – .22 • Do these (to 2 decimal places): • x = log 6 • 2. x = log 0.03 • 3. x = ln 104 • 4. x = ln 0.8 You should check your answers! * We will learn how to use the calculator for bases other than10 and e later.

  9. 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

  10. 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: • log 500 = x • ex = e5 • ex = 200 • log 1000 = x • Solve the above problems (which ones need a calculator?)

  11. 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 • 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 x = by↔ y = logb x

  12. Wrap Up (day 1) or Warm Up (day 2) log 8 = 0.9031 → 100.9031 = 8 10???? = 80 ← log 80 100.9031 ≈ 8 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. 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

  13. 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, dB = , where I0 is the intensity of a barely audible sound. [Page 192 has a table with intensity levels for common sounds.] * There are many others: brightness, acidity, & earthquake magnitude. • Examples: • Average car @ 70kmh, I = 106.8I0. dB? • 2. Whisper, I = 101.5I0. dB of 2 people whispering? I person whispering: 15dB 2 people 18dB You do not have to memorize these defintions.

  14. 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: 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. • 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?

  15. joules joules times more powerful Richter Scale: An Application of Logs • 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

  16. 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· e2use rules of exponents y = e2 · (e4)x use rules of exponents y ≈ 7.4(54.6)xestimate e2 and e4

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