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**Unformatted text preview: **MAC 1140
Module 8 Logarithmic Functions Rev.S08 Learning Objectives 1. 2. 3. 4. 5. 6. 7. 8. Upon completing this module, you should be able to evaluate the common logarithmic function. solve basic exponential and logarithmic equations. evaluate logarithms with other bases. solve general exponential and logarithmic equations. apply basic properties of logarithms. use the change of base formula. solve exponential equations. solve logarithmic equations. Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 2 Logarithmic Functions
There are three sections in this module: 5.4 5.5 5.6 Logarithmic Functions and Models Properties of Logarithms Exponential and Logarithmic Equations Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 3 What is the Definition of a Common Logarithmic Function? The common logarithm of a positive number x, denoted log (x), is defined by log (x) = k if and only if x = 10k where k is a real number. The function given by f(x) = log (x) is called the common logarithmic function. Note that the input x must be positive. Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 4 Let's Evaluate Some Common Logarithms log (10) log (100) log (1000) log (10000) log (1/10) log (1/100) log (1/1000) log (1)
Rev.S08 1 because 101 = 10 2 because 102 = 100 3 because 103 = 1000 4 because 104 = 10000 1 because 10-1 = 1/10 2 because 10-2 = 1/100 3 because 10-3 = 1/1000 0 because 100 = 1
http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 5 Let's Take a Look at the Graph of a Logarithmic Function
x f(x) 0.01 0.1 1 10 100
Rev.S08 -2 -1 0 1 2
Note that the graph of y = log (x) is the graph of y = 10x reflected through the line y = x. This suggests that these are inverse functions.
http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 6 What is the Inverse Function of a Common Logarithmic Function? Note that the graph of f(x) = log (x) passes the horizontal line test so it is a one-to-one function and has an inverse function. Find the inverse of y = log (x) Using the definition of common logarithm to solve for x gives x = 10y Interchanging x and y gives y = 10x Thus, the inverse of y = log (x) is y = 10x Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 7 What is the Inverse Properties of the Common Logarithmic Function? Recall that f -1(x) = 10x given f(x) = log (x) Since (f f -1 )(x) = x for every x in the domain of f -1 log(10x) = x for all real numbers x. log( Since (f -1 f)(x) = x for every x in the domain of f logx 10logx = x for any positive number x Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 8 What is the Definition of a Logarithmic Function with base a? The logarithm with base a of a positive number x, denoted by loga(x) is defined by loga(x) = k if and only if x = ak where a > 0, a 1, and k is a real number. The function given by f(x) = loga(x) is called the logarithmic function with base a. Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 9 What is the Natural Logarithmic Function? Logarithmic Functions with Base 10 are called "common logs." log (x) means log10(x) - The Common Logarithmic Function Logarithmic Functions with Base e are called "natural logs." ln (x) means loge(x) - The Natural Logarithmic Function Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 10 Let's Evaluate Some Natural Logarithms ln (e) ln (e 2 ) ln (e) = loge(e) = 1 since e1= e ln(e2) = loge (e2) = 2 since 2 is the exponent that goes on e to produce e2 . ln (1) = loge1 = 0 since e0= 1 1/2 since 1/2 is the exponent that goes on e to produce e1/2
Rev.S08
http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. ln (1) 11 What is the Inverse of a Logarithmic Function with base a? Note that the graph of f(x) = loga(x) passes the horizontal line test so it is a one-to-one function and has an inverse function. Find the inverse of y = loga(x) Using the definition of common logarithm to solve for x gives x = ay Interchanging x and y gives y = ax Thus, the inverse of y = loga(x) is y = ax
Rev.S08
http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 12 What is the Inverse Properties of a Logarithmic Function with base a? Recall that f -1(x) = ax given f(x) = loga(x) Since (f f -1 )(x) = x for every x in the domain of f -1 loga(ax) = x for all real numbers x. Since (f -1 f)(x) = x for every x in the domain of f alogax = x for any positive number x Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 13 Let's Try to Solve Some Exponential Equations Solve the equation 4x = 1/64 Take the log of both sides to the base 4 log4 (4x) = log4(1/64) Using the inverse property loga (ax) =x , this simplifies to x = log4(1/64) Since 1/64 can be rewritten as 43 x = log4(43) Using the inverse property loga (ax) = x , this simplifies to x = 3
Rev.S08
http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 14 Let's Try to Solve Some Exponential Equations (Cont.) Solve the equation ex = 15 Take the log of both sides to the base e ln(ex) = ln(15) Using the inverse property loga(ax) = x this simplifies to x = ln(15) Using the calculator to estimate ln (15) x 2.71 Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 15 Let's Try to Solve Some Logarithmic Equations (Cont.) Solve the equation ln (x) = 1.5 Exponentiate both sides using base e lnx elnx = e1.5 Using the inverse property alogax = x this simplifies to x = e1.5 Using the calculator to estimate e1.5 x 4.48 Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 16 What are the Basic Properties of Logarithms? Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 17 Property 1 loga(1) = 0 and loga(a) = 1 a0 =1 and a1 = a Note that this property is a direct result of the inverse property loga(ax) = x Example: log (1) =0 and ln (e) =1 Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 18 Property 2 loga(m) + loga(n) = loga(mn) The sum of logs is the log of the product. Example: Let a = 2, m = 4 and n = 8 loga(m) + loga(n) = log2(4) + log2(8) = 2 + 3 loga(mn) = log2(4 8) = log2(32) = 5 Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 19 Property 3 The difference of logs is the log of the quotient. Example: Let a = 2, m = 4 and n = 8 Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 20 Property 4 Example: Let a = 2, m = 4 and r = 3 Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 21 Example Expand the expression. Write without exponents. Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 22 One More Example Write as the logarithm of a single expression Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 23 What is the Change of Base Formula? Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 24 Example of Using the Change of Base Formula? Use the change of base formula to evaluate log38 Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 25 Modeling Compound Interest How long does it take money to grow from $100 to $200 if invested into an account which compounds quarterly at an annual rate of 5%? Must solve for t in the following equation Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 26 Modeling Compound Interest (Cont.) Divide each side by 100 Take common logarithm of each side Property 4: log(mr) = r log (m) Divide each side by 4log1.0125 Approximate using calculator Rev.S08
http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 27 Modeling Compound Interest (Cont.)
Alternatively, we can take natural logarithm of each side instead of taking the common logarithm of each side.
Divide each side by 100 Take natural logarithm of each side Property 4: ln (mr) = r ln (m) Divide each side by 4 ln (1.0125) Approximate using calculator Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 28 Solve 3(1.2)x + 2 = 15 for x symbolically
Divide each side by 3 Take common logarithm of each side (Could use natural logarithm) Property 4: log(mr) = r log (m) Divide each side by log (1.2) Approximate using calculator
http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08 29 Solve ex+2 = 52x for x symbolically
Take natural logarithm of each side Property 4: ln (mr) = r ln (m) ln (e) = 1 Subtract 2x ln(5) and 2 from each side Factor x from left-hand side Divide each side by 1 2 ln (5) Approximate using calculator Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 30 Solving a Logarithmic Equation Symbolically In developing countries there is a relationship between the amount of land a person owns and the average daily calories consumed. This relationship is modeled by the formula C(x) = 280 ln(x+1) + 1925 where x is the amount of land owned in acres and
Source: D. Gregg: The World Food Problem Determine the number of acres owned by someone whose average intake is 2400 calories per day. Must solve for x in the equation 280 ln(x+1) + 1925 = 2400
Rev.S08
http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 31 Solving a Logarithmic Equation Symbolically (Cont.)
Subtract 1925 from each side Divide each side by 280 Exponentiate each side base e
lnk Inverse property elnk = k Subtract 1 from each side Approximate using calculator Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 32 One More Example Definition of logarithm logax = k if and only if x = ak Add x to both sides of equation Subtract 2 from both sides of the equation Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 33 What have we learned? 1. 2. 3. 4. 5. 6. 7. 8. We have learned to evaluate the common logarithmic function. solve basic exponential and logarithmic equations. evaluate logarithms with other bases. solve general exponential and logarithmic equations. apply basic properties of logarithms. use the change of base formula. solve exponential equations. solve logarithmic equations. Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 34 Credit Some of these slides have been adapted/modified in part/whole from the slides of the following textbook: Rockswold, Gary, Precalculus with Modeling and Visualization, 3th Edition Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 35 ...

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