TERRA Environmental Research Institute



Name:____________________ Per:_____ Revised 2015- Assignment # 6Lessons: Section 7.3 Logarithmic Functions as Inverses EXPONENTIAL NOTATION VS LOGARITHMIC NOTATIONRead table carefully then fill in the blank boxesExponential formLogarithmic FormNOTES23 = 8 log28=3 Read as “ log base 2 of 8 is 3”notice that log is the exponent!!3 is the exponent to which 2 must be raised to get 83-2= 19 log3(19)= -2Read as “log base 3 of-2 is (19)”52 = 25 And it is read as :log 3 81 = 4 104=10000 log 10000=4 Read as “ log 1000 is 4”No need to write the base when it is 10when the base is 10, the base is explicit and it is called a Common log log10 1000 → log 1000e2 ≈7.4 ln 7.4≈ 2 Read as “the natural log of 7.4 is close to 2”when the base is e is called a Natural log, loge 7.4 → ln 7.4 Read carefully and fill in the blanksNote: Look at your scientific calculator, you have a log key and a ln key, therefore your calculator can compute ONLY logs with base 10 (common logs) or base e (natural logs). Let’s check problems above. Use your calculator and enter log 10000, did you get 4? _____ Enter ln 7.4, did you get ≈2?______ How do you compute log28 or log3(19) using a calculator? These two logs have base 2 and base 3!! To be able to compute logs of other bases using a calculator, you must use the Change of Base formula on p. 464 We will tackle this issue later on.Essential Understanding : The INVERSE of the exponential function y = bx is x =______ and it is a function. To express this INVERSE in terms of y ( “y as a function of x”), write it using log notation → y = _________Definition of Logarithm base b for positive number x satisfies the following definition: (p.451) Logb ___= ___ if and only if ____________Where b>____ , b≠____ and x>0 ( only defined for positive values of x, why? _____________________________ )The domain of this function is x >0 and based on these x-values, the Range of this Function is y ∈______________ ( think of it as an exponential → x = by can y be negative, zero, positive? Can y be any Real number?)1. Graph [Exponential function : y = 10x ] , [ logarithmic function : y = log x ] and [the diagonal line y=x (in RED)] on the same grid. Show graphically that y=10x and y =log x are inverse functions . Fill in table of values . Label graphs. If needed, use your scientific calculator to you fill in the table of values. Pay attention to domain of function. Graph & label Exponential, Logarithmic and RED Line y=10xy = log xx Y xY-1010-10=-1Error why?-810-8=0Error why?-50.00000000001-30.0000001-20.1-110-1=1Log 1 = 0100=20.551101=1010Log 10 =y = log x → Domain:_______ Range:_______ Y = 10x → Domain:_______________ Range:_______________Read Concept Summary page 435. All transformations together y= a logb (x – h) + k2. On the same grid, graph the parent graph y = log4x (highlight it), graph y=log4(x+3) label it, and y=log4x +3 label it. Hint: y = log4x → 4y= x, use the exponential equation to find points for your graph. ( Similar to example on p.455)y=log4(x+3) → 4y= (x +3), solving for x we get x = 4y -3 and y=log4x +3 → y-3 = log4x → 4(y -3)= x (tricky!) Graph and LABEL all 3 graphs y = log4x y=log4(x+3) y=log4x +34y= xy4y – 3 = xy4(y-3)= xy-3-3-3-2-2-2-1-1-100011122233344410001252171700 =86614024955500Recall log b x = y if and only if by = x b Raised to y equals x Log lingo→ b is the base, x is called the argument3)Simplify each expression. Show work in detail and neatly written. No calculators!!( See example 2 on p. 452)a) log2 32= ____ Work→b) log4 32= ____ Work→c) log64 132= ____ Work→d) log2 2 = ____ Work→e) log (11000) = _____ Work→f) ln (1e) = _______ Work→g) log 16 (x) = 16 ( write your answer in simplest radical form) Work→h) log 3 981 = _______ ( answer is a fraction) Work→log66=log77=log88 = ___ Work→j) log13127 = ____________ Work→4) Write in exponential form. a) log 3 = x ________ b) ln 5 = A __________ c) log3 7 = x + 3 _____________d) log3 7 + 2 = x (hint; isolate log then convert to expo. Form) _____________________________ e) log 6 – 1 = x ______________________________________________________________5) True or False. Explain, show work in detail.NWNC!! a) log2 48 = 34 __________________________________________ b) log 100 = log22 ______________________ c) log3 (0.037037037…) = -3 ( Hint: express the repeating decimal as a fraction)d) logaa=logbb ________________________________________________________________e) 2x = a + 1 in log form is log2 a + 1 = x, this is false, how can you write it so it is true? ________________Do the Got it problems following each example + 10, 11,26,28,33,35,42, 43,44,58,59, 61, 67,68,75,76,79,80-83Start Assignment # 7 Got it problems on Section 7.4 ( use a different sheet of paper since it is a different assignment) ................
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