# OSU MTH 111 - Act-#17 Log Ftns-Key (6 pages)

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## Act-#17 Log Ftns-Key

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- School:
- Oregon State University
- Course:
- Mth 111 - College Algebra

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Name Key Grp Math 111 Logarithmic Functions GrpAct 17 Prerequisite Skills Key Terms working with inverse functions composing functions with their inverse working with exponential functions Logarithmic Function Natural Logarithm Common Logarithm Logarithm Properties Inverse Properties for Logs Change of Base Property Exact Approximate solutions Sects 5 4 5 5 Learning Outcomes Convert between exponential and logarithmic statements Evaluate logarithmic expressions Apply change of base formula to logarithmic expressions Use logarithms to solve exponential equations Use exponentiation to solve logarithmic equations Determine domain range of logarithmic functions Apply the properties of logarithms to simplify or expand logarithmic expressions Model with logarithmic functions x 1 1 Last class we talked about exponential functions Let f x 4 x and g x Evaluate each of the following 5 1 25 d g 2 25 a g 2 b f 0 5 2 c f 3 e f 0 1 f g 0 1 1 64 p is the Power that goes on the Base b to give you m 2 This class we ll look at logarithmic functions In your MML HW and Reading you saw how exponential and logarithmic equations are related Fill in the table with the alternate forms of each equation Exponential Form Logarithmic Form Written Form 1 log 5 2 25 The base 5 to the 2 power is equal to 23 8 log 2 8 3 The base 2 to the 3 power is equal to 8 43 64 log 4 64 3 811 4 3 log 81 3 5 2 7 1 1 25 1 7 1001 2 10 The base 4 to the 3 power is equal to 64 1 4 1 log 7 1 7 log100 10 1 2 130 1 log13 1 0 by x log b x y e3 4657 32 1 25 ln 32 3 4657 see note below The base 81 to the 1 power is equal to 3 4 The base 7 to the 1 power is equal to The base 100 to the 1 7 1 power is equal to 10 2 The base 13 to the 0 power is equal to 1 The base b to the y power is equal to x The base e to the 3 4657 power is equal to 32 Don t know what ln means It is the logarithm with base e ln x loge x This logarithm is used a lot in mathematics and it is called the natural logarithm Do you remember last week when we saw exponential functions with base e Those were called natural exponential functions 3 The domain and range of logarithmic and exponential functions x a What are the domain and range of f x b D R 0 You may have already realized this but If then b Explain why the equation ex 0 does not have any solutions A positive based raised to any power is greater than zero The inverse function of an exponential function is a logarithmic function e x 0 e 01 x 0 BUT e 0 therefore e x 0 for any value of x c Try to evaluate ln 0 Explain Remember if m e p then p ln m ln 0 z e z 0 So by part b ln 0 Does Not Exist DNE d Explain why the equation 5x 15 does not have any solutions A positive based raised to any power is greater than zero e Try to evaluate log5 15 Explain Remember if m 5 p then p log 5 m log 5 15 z 5z 15 So by part d log 5 15 Does Not Exist DNE f What is the domain and range of logb x The function logb x has the same domain and range regardless of the base b D 0 R 4 On the axes below sketch a graph of y ln x and y log5 x Indicate which graph is which Start by plotting a couple of points on each graph Label the x intercept s on each graph Do NOT use your calculator to get the graph x 1 1 2 5 2 25 5 y log5 x x y ln x 1 log 5 log 5 5 2 2 25 1 e 2 2 e 1 ln 2 ln e 2 2 e 1 5 1 5 1 log 5 log 5 5 1 1 5 1 e 1 e 1 ln ln e 1 1 e 1 50 log 5 1 log 5 50 0 1 e0 ln 1 ln e0 0 5 51 log 5 51 1 e e1 ln e ln e1 1 52 log 5 52 2 e2 ln e2 2 53 log 5 53 3 e3 ln e3 3 5 Evaluate each logarithm and write down a written statement of what it means Recall y log b x y is the power that goes on the base b to give you x a log 3 81 4 c ln e9 9 b 1 log 2 100 d log3 3 7 7 Natural and Common Logarithms loge x is called the natural logarithm and is written ln x log10 x is called the common logarithm and is written log x 6 Using Logarithms to solve Exponential Equations The box at the right gives two very important properties of logarithms If you look back at the previous problem you ll see you were actually using the first inverse property on parts c and d That first inverse rule gives us a way to solve exponential equations Use it to solve the equations below give the Exact solution for each and the approximate solutions if possible Inverse properties of logarithms a 10 210 x x log 210 2 3222 Exact and Approximate values x b e 21 5 is an exact value is an approximate value x ln 105 4 6540 is an exact value is an c approximate value 52 x 13 Note that some logarithms can be simplified exactly For example log3 9 2 is an exact value 1 ln 13 x 0 7968 2 ln 5 7 Using exponentials to solve logarithmic equations To solve a logarithmic equation we use the property in the box below and the other inverse property stated in the box near the top of the page Check your solutions a log 5 x 2 x 20 Using Exponentiation to Solve an Equation If then for The above statement is true because when we input equivalent quantities into a function the resulting outputs are also equal b 3ln x 12 x e 4 54 5982 By carefully selecting the value of b we can use this idea to solve a logarithmic equation like those to the left After exponentiating both sides we can use the inverse rule 8 Properties of Logarithms We already saw the important inverse properties of logarithms but there are a couple of other really important properties of logarithms given in the table at the right Your friend is trying to understand these properties and she is not convinced that they are true It is your job to demonstrate for your friend that they are indeed true Using only the calculations in the …

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