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BMayer@ChabotCollege.edu MTH15_Lec-19_sec_4-2_Logarithmic_Fcns.pptx 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics §4.2 Log Functions

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BMayer@ChabotCollege.edu MTH15_Lec-19_sec_4-2_Logarithmic_Fcns.pptx 2 Bruce Mayer, PE Chabot College Mathematics Review § Any QUESTIONS About §4.1 → Exponential Functions Any QUESTIONS About HomeWork §4.1 → HW-18 4.1

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BMayer@ChabotCollege.edu MTH15_Lec-19_sec_4-2_Logarithmic_Fcns.pptx 3 Bruce Mayer, PE Chabot College Mathematics §4.2 Learning Goals Define and explore logarithmic functions and their properties Use logarithms to solve exponential equations Examine applications involving logarithms John Napier (1550-1617) Logarithm Pioneer

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BMayer@ChabotCollege.edu MTH15_Lec-19_sec_4-2_Logarithmic_Fcns.pptx 4 Bruce Mayer, PE Chabot College Mathematics Logarithm → What is it? Concept: If b > 0 and b ≠ 1, then y = log b x is equivalent to x = b y Symbolically x = b y y = log b x The exponent is the logarithm. The base is the base of the logarithm.

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BMayer@ChabotCollege.edu MTH15_Lec-19_sec_4-2_Logarithmic_Fcns.pptx 5 Bruce Mayer, PE Chabot College Mathematics Logarithm Illustrated Consider the exponential function y = f(x) = 3 x. Like all exponential functions, f is one-to-one. Can a formula for the inverse Function, x = g(y) be found? f − 1 (x) ≡ the exponent to which we must raise 3 to get x. y = 3 x x = 3 y y ≡ the exponent to which we must raise 3 to get x. Need

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BMayer@ChabotCollege.edu MTH15_Lec-19_sec_4-2_Logarithmic_Fcns.pptx 6 Bruce Mayer, PE Chabot College Mathematics Logarithm Illustrated Now define a new symbol to replace the words “the exponent to which we must raise 3 to get x”: log 3 x, read “the logarithm, base 3, of x,” or “log, base 3, of x,” means “the exponent to which we raise 3 to get x.” Thus if f(x) = 3 x, then f −1 (x) = log 3 x. Note that f −1 (9) = log 3 9 = 2, as 2 is the exponent to which we raise 3 to get 9

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BMayer@ChabotCollege.edu MTH15_Lec-19_sec_4-2_Logarithmic_Fcns.pptx 7 Bruce Mayer, PE Chabot College Mathematics Example Evaluate Logarithms Evaluate: a) log 3 81 b) log 3 1 c) log 3 (1/9) Solution: a)Think of log 3 81 as the exponent to which we raise 3 to get 81. The exponent is 4. Thus, since 3 4 = 81, log 3 81 = 4. b)ask: “To what exponent do we raise 3 in order to get 1?” That exponent is 0. So, log 3 1 = 0 c)To what exponent do we raise 3 in order to get 1/9? Since 3 −2 = 1/9 we have log 3 (1/9) = −2

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BMayer@ChabotCollege.edu MTH15_Lec-19_sec_4-2_Logarithmic_Fcns.pptx 8 Bruce Mayer, PE Chabot College Mathematics The Meaning of log a x For x > 0 and a a positive constant other than 1, log a x is the exponent to which a must be raised in order to get x. Thus, log a x = m means a m = x or equivalently, log a x is that unique exponent for which

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BMayer@ChabotCollege.edu MTH15_Lec-19_sec_4-2_Logarithmic_Fcns.pptx 9 Bruce Mayer, PE Chabot College Mathematics Example Exponential to Log Write each exponential equation in logarithmic form. Soln

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BMayer@ChabotCollege.edu MTH15_Lec-19_sec_4-2_Logarithmic_Fcns.pptx 10 Bruce Mayer, PE Chabot College Mathematics Example Log to Exponential Write each logarithmic equation in exponential form Soln

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BMayer@ChabotCollege.edu MTH15_Lec-19_sec_4-2_Logarithmic_Fcns.pptx 11 Bruce Mayer, PE Chabot College Mathematics Example Evaluate Logarithms Find the value of each of the following logarithms Solution

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BMayer@ChabotCollege.edu MTH15_Lec-19_sec_4-2_Logarithmic_Fcns.pptx 12 Bruce Mayer, PE Chabot College Mathematics Example Evaluate Logarithms Solution (cont.)

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BMayer@ChabotCollege.edu MTH15_Lec-19_sec_4-2_Logarithmic_Fcns.pptx 13 Bruce Mayer, PE Chabot College Mathematics Example Use Log Definition Solve each equation for x, y or z Solution

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BMayer@ChabotCollege.edu MTH15_Lec-19_sec_4-2_Logarithmic_Fcns.pptx 14 Bruce Mayer, PE Chabot College Mathematics Example Use Log Definition Solution (cont.)

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BMayer@ChabotCollege.edu MTH15_Lec-19_sec_4-2_Logarithmic_Fcns.pptx 15 Bruce Mayer, PE Chabot College Mathematics Inverse Property of Logarithms Recall Def: For x > 0, a > 0, and a ≠ 1, In other words, The logarithmic function is the inverse function of the exponential function; e.g.

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BMayer@ChabotCollege.edu MTH15_Lec-19_sec_4-2_Logarithmic_Fcns.pptx 16 Bruce Mayer, PE Chabot College Mathematics Derive Change of Base Rule Any number >1 can be used for b, but since most calculators have ln and log functions we usually change between base-e and base-10

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BMayer@ChabotCollege.edu MTH15_Lec-19_sec_4-2_Logarithmic_Fcns.pptx 17 Bruce Mayer, PE Chabot College Mathematics Example Inverse Property Evaluate: Solution Remember that log 5 23 is the exponent to which 5 is raised to get 23. Raising 5 to that exponent, obtain

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BMayer@ChabotCollege.edu MTH15_Lec-19_sec_4-2_Logarithmic_Fcns.pptx 18 Bruce Mayer, PE Chabot College Mathematics Basic Properties of Logarithms For any base a > 0, with a ≠ 1, Discern from the Log Definition 1.Log a a = 1 As 1 is the exponent to which a must be raised to obtain a (a 1 = a) 2.Log a 1 = 0 As 0 is the exponent to which a must be raised to obtain 1 (a 0 = 1)

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BMayer@ChabotCollege.edu MTH15_Lec-19_sec_4-2_Logarithmic_Fcns.pptx 19 Bruce Mayer, PE Chabot College Mathematics Graph Logarithmic Function Sketch the graph of y = log 3 x Soln: Make T-Table → xy = log 3 x(x, y) 3 –3 = 1/27–3(1/27, –3) 3 –2 = 1/9–2(1/9, –2) 3 –3 = 1/3–1(1/3, –1) 3 0 = 10(1, 0) 3 1 = 31(3, 1) 3 2 = 92(9, 2)

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BMayer@ChabotCollege.edu MTH15_Lec-19_sec_4-2_Logarithmic_Fcns.pptx 20 Bruce Mayer, PE Chabot College Mathematics Graph Logarithmic Function Plot the ordered pairs and connect the dots with a smooth curve to obtain the graph of y = log 3 x

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BMayer@ChabotCollege.edu MTH15_Lec-19_sec_4-2_Logarithmic_Fcns.pptx 21 Bruce Mayer, PE Chabot College Mathematics Example Graph by Inverse Graph y = f(x) = 3 x Solution: Use Inverse Relation for Logs & Exponentials Reflect the graph of y = 3 x across the line y = x to obtain the graph of y = f −1 (x) = log 3 x

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BMayer@ChabotCollege.edu MTH15_Lec-19_sec_4-2_Logarithmic_Fcns.pptx 22 Bruce Mayer, PE Chabot College Mathematics Properties of Exponential and Logarithmic Functions Exponential Function f (x) = a x Logarithmic Function f (x) = log a x Domain (0, ∞) Range (–∞, ∞) Domain (–∞, ∞) Range (0, ∞) x-intercept is 1 No y-intercept y-intercept is 1 No x-intercept x-axis (y = 0) is the horizontal asymptote y-axis (x = 0) is the vertical asymptote

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BMayer@ChabotCollege.edu MTH15_Lec-19_sec_4-2_Logarithmic_Fcns.pptx 23 Bruce Mayer, PE Chabot College Mathematics Properties of Exponential and Logarithmic Functions Exponential Function f (x) = a x Logarithmic Function f (x) = log a x Is one-to-one, that is, log a u = log a v if and only if u = v Is one-to-one, that is, a u = a v if and only if u = v Increasing if a > 1 Decreasing if 0 < a < 1

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BMayer@ChabotCollege.edu MTH15_Lec-19_sec_4-2_Logarithmic_Fcns.pptx 24 Bruce Mayer, PE Chabot College Mathematics Graphs of Logarithmic Fcns f (x) = log a x (0 < a < 1)f (x) = log a x (a > 1)

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BMayer@ChabotCollege.edu MTH15_Lec-19_sec_4-2_Logarithmic_Fcns.pptx 25 Bruce Mayer, PE Chabot College Mathematics Common Logarithms The logarithm with base 10 is called the common logarithm and is denoted by omitting the base: logx = log 10 x. So y = logx if and only if x = 10 y Applying the basic properties of logs 1.log(10) = 1 2.log(1) = 0 3.log(10 x ) = x 4.10 logx = x

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BMayer@ChabotCollege.edu MTH15_Lec-19_sec_4-2_Logarithmic_Fcns.pptx 26 Bruce Mayer, PE Chabot College Mathematics Common Log Convention By this Mathematics CONVENTION the abbreviation log, with no base written, is understood to mean logarithm base 10, or a common logarithm. Thus, log21 = log 10 21 On most calculators, the key for common logarithms is marked LOG

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BMayer@ChabotCollege.edu MTH15_Lec-19_sec_4-2_Logarithmic_Fcns.pptx 27 Bruce Mayer, PE Chabot College Mathematics Natural Logarithms Logarithms to the base “e” are called natural logarithms, or Napierian logarithms, in honor of John Napier, who first “discovered” logarithms. The abbreviation “ln” is generally used with natural logarithms. Thus, ln 21 = log e 21. On most calculators, the key for natural logarithms is marked LN

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BMayer@ChabotCollege.edu MTH15_Lec-19_sec_4-2_Logarithmic_Fcns.pptx 28 Bruce Mayer, PE Chabot College Mathematics Natural Logarithms The logarithm with base e is called the natural logarithm and is denoted by ln x. That is, ln x = log e x. So y = lnx if and only if x = e y Applying the basic properties of logs 1.ln(e) = 1 2.ln(1) = 0 3.ln(e x ) = x 4.e lnx = x

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BMayer@ChabotCollege.edu MTH15_Lec-19_sec_4-2_Logarithmic_Fcns.pptx 29 Bruce Mayer, PE Chabot College Mathematics Example Sound Intensity This function is sometimes used to calculate sound intensity Where d ≡ the intensity in decibels, I ≡ the intensity watts per unit of area I 0 ≡ the faintest audible sound to the average human ear, which is 10 −12 watts per square meter (1x10 −12 W/m 2 ).

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BMayer@ChabotCollege.edu MTH15_Lec-19_sec_4-2_Logarithmic_Fcns.pptx 30 Bruce Mayer, PE Chabot College Mathematics Example Sound Intensity Use the Sound Intensity Equation (a.k.a. the “dBA” Eqn) to find the intensity level of sounds at a decibel level of 75 dB? Solution: We need to isolate the intensity, I, in the dBA eqn

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BMayer@ChabotCollege.edu MTH15_Lec-19_sec_4-2_Logarithmic_Fcns.pptx 31 Bruce Mayer, PE Chabot College Mathematics Example Sound Intensity Solution (cont.) in the dBA eqn substitute 75 for d and 10 −12 for I 0 and then solve for I

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BMayer@ChabotCollege.edu MTH15_Lec-19_sec_4-2_Logarithmic_Fcns.pptx 32 Bruce Mayer, PE Chabot College Mathematics Example Sound Intensity Thus the Sound Intensity at 75 dB is 10 −4.5 W/m 2 = 10 −9/2 W/m 2 Using a Scientific calculator and find that I = 3.162x10 −5 W/m 2 = 31.6 µW/m 2

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BMayer@ChabotCollege.edu MTH15_Lec-19_sec_4-2_Logarithmic_Fcns.pptx 33 Bruce Mayer, PE Chabot College Mathematics Example Sound Intensity Check If the sound intensity is 10 −4.5 W/m 2, verify that the decibel reading is 75.

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BMayer@ChabotCollege.edu MTH15_Lec-19_sec_4-2_Logarithmic_Fcns.pptx 34 Bruce Mayer, PE Chabot College Mathematics Summary of Log Rules For any positive numbers M, N, and a with a ≠ 1, and whole number p Product Rule Power Rule Quotient Rule Base-to-Power Rule

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BMayer@ChabotCollege.edu MTH15_Lec-19_sec_4-2_Logarithmic_Fcns.pptx 35 Bruce Mayer, PE Chabot College Mathematics Typical Log-Confusion Beware Beware that Logs do NOT behave Algebraically. In General:

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BMayer@ChabotCollege.edu MTH15_Lec-19_sec_4-2_Logarithmic_Fcns.pptx 36 Bruce Mayer, PE Chabot College Mathematics Change of Base Rule Let a, b, and c be positive real numbers with a ≠ 1 and b ≠ 1. Then log b x can be converted to a different base as follows:

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BMayer@ChabotCollege.edu MTH15_Lec-19_sec_4-2_Logarithmic_Fcns.pptx 37 Bruce Mayer, PE Chabot College Mathematics Derive Change of Base Rule Any number >1 can be used for b, but since most calculators have ln and log functions we usually change between base-e and base-10

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BMayer@ChabotCollege.edu MTH15_Lec-19_sec_4-2_Logarithmic_Fcns.pptx 38 Bruce Mayer, PE Chabot College Mathematics Example Evaluate Logs Compute log 5 13 by changing to (a) common logarithms (b) natural logarithms Soln

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BMayer@ChabotCollege.edu MTH15_Lec-19_sec_4-2_Logarithmic_Fcns.pptx 39 Bruce Mayer, PE Chabot College Mathematics Use the change-of-base formula to calculate log 5 12. Round the answer to four decimal places Solution Example Evaluate Logs Check

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BMayer@ChabotCollege.edu MTH15_Lec-19_sec_4-2_Logarithmic_Fcns.pptx 40 Bruce Mayer, PE Chabot College Mathematics Find log 3 7 using the change-of-base formula Solution Example Evaluate Logs Substituting into

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BMayer@ChabotCollege.edu MTH15_Lec-19_sec_4-2_Logarithmic_Fcns.pptx 41 Bruce Mayer, PE Chabot College Mathematics Example Use The Rules Express as an equivalent expression using individual logarithms of x, y, & z Soln a) = log 4 x 3 – log 4 yz = 3log 4 x – log 4 yz = 3log 4 x – (log 4 y + log 4 z) = 3log 4 x – log 4 y – log 4 z

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BMayer@ChabotCollege.edu MTH15_Lec-19_sec_4-2_Logarithmic_Fcns.pptx 42 Bruce Mayer, PE Chabot College Mathematics Example Use The Rules Soln b)

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BMayer@ChabotCollege.edu MTH15_Lec-19_sec_4-2_Logarithmic_Fcns.pptx 43 Bruce Mayer, PE Chabot College Mathematics Caveat on Log Rules Because the product and quotient rules replace one term with two, it is often best to use the rules within parentheses, as in the previous example

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BMayer@ChabotCollege.edu MTH15_Lec-19_sec_4-2_Logarithmic_Fcns.pptx 44 Bruce Mayer, PE Chabot College Mathematics Example Cesium-137 ½-Life A sample of radioactive Cesium-137 has been Stored, unused, for cancer treatment for 2.2 years. In that time, 5% of the original sample has decayed. What is the half-life (time required to reduce the radioactive substance to half of its starting quantity) of Cesium-137?

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BMayer@ChabotCollege.edu MTH15_Lec-19_sec_4-2_Logarithmic_Fcns.pptx 45 Bruce Mayer, PE Chabot College Mathematics Example Cesium-137 ½-Life SOLUTION: Start with the math model for exponential Decay Recall the Given information: after 2.2 years, 95% of the sample remains Use the Model and given data to find k Use data in Model: Divide both sides by A 0 :

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BMayer@ChabotCollege.edu MTH15_Lec-19_sec_4-2_Logarithmic_Fcns.pptx 46 Bruce Mayer, PE Chabot College Mathematics Example Cesium-137 ½-Life Now take the ln of both Sides Using the Base-to-Power Rule Find by Algebra Now set the amount, A, to ½ of A 0

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BMayer@ChabotCollege.edu MTH15_Lec-19_sec_4-2_Logarithmic_Fcns.pptx 47 Bruce Mayer, PE Chabot College Mathematics Example Cesium-137 ½-Life After dividing both sides by A 0 Taking the ln of Both Sides Solving for the HalfLife State: The HalfLife of Cesion-137 is approximately 29.7 years

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BMayer@ChabotCollege.edu MTH15_Lec-19_sec_4-2_Logarithmic_Fcns.pptx 48 Bruce Mayer, PE Chabot College Mathematics Example Compound Interest In a Bank Account that Compounds CONTINUOUSLY the relationship between the $-Principal, P, deposited, the Interest rate, r, the Compounding time-period, t, and the $-Amount, A, in the Account:

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BMayer@ChabotCollege.edu MTH15_Lec-19_sec_4-2_Logarithmic_Fcns.pptx 49 Bruce Mayer, PE Chabot College Mathematics Example Compound Interest If an account pays 8% annual interest, compounded continuously, how long will it take a deposit of $25,000 to produce an account balance of $100,000? Familiarize In the Compounding Eqn replace P with 25,000, r with 0.08, A with $100,000, and then simplify.

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BMayer@ChabotCollege.edu MTH15_Lec-19_sec_4-2_Logarithmic_Fcns.pptx 50 Bruce Mayer, PE Chabot College Mathematics Example Compound Interest Solution Substitute. Divide. Approximate using a calculator. State Answer The account balance will reach $100,000 in about 17.33 years.

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BMayer@ChabotCollege.edu MTH15_Lec-19_sec_4-2_Logarithmic_Fcns.pptx 51 Bruce Mayer, PE Chabot College Mathematics Example Compound Interest Check: Because 17.33 was not the exact time, $100,007.45 is reasonable for the Chk

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BMayer@ChabotCollege.edu MTH15_Lec-19_sec_4-2_Logarithmic_Fcns.pptx 52 Bruce Mayer, PE Chabot College Mathematics WhiteBoard Work Problems From §4.2 P72 → Atmospheric Pressure at Altitude –See also: B. Mayer, “Small Signal Analysis of Source Vapor Control Requirements for APCVD”, IEEE Transactions on Semiconductor Manufacturing, vol. 9, no. 3, 1996, pg 355

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BMayer@ChabotCollege.edu MTH15_Lec-19_sec_4-2_Logarithmic_Fcns.pptx 53 Bruce Mayer, PE Chabot College Mathematics All Done for Today Napier’s MasterWork Year 1619

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BMayer@ChabotCollege.edu MTH15_Lec-19_sec_4-2_Logarithmic_Fcns.pptx 54 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics Appendix –

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BMayer@ChabotCollege.edu MTH15_Lec-19_sec_4-2_Logarithmic_Fcns.pptx 55 Bruce Mayer, PE Chabot College Mathematics ConCavity Sign Chart abc −−−−−−++++++−−−−−−++++++ x ConCavity Form d 2 f/dx 2 Sign Critical (Break) Points InflectionNO Inflection Inflection

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BMayer@ChabotCollege.edu MTH15_Lec-19_sec_4-2_Logarithmic_Fcns.pptx 56 Bruce Mayer, PE Chabot College Mathematics

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BMayer@ChabotCollege.edu MTH15_Lec-19_sec_4-2_Logarithmic_Fcns.pptx 57 Bruce Mayer, PE Chabot College Mathematics

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BMayer@ChabotCollege.edu MTH15_Lec-19_sec_4-2_Logarithmic_Fcns.pptx 58 Bruce Mayer, PE Chabot College Mathematics

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