Unit 7: Exponential and Logarithmic Functions Lesson #2: Exponential Functions and Their 1. Graph 2 by making a table. x f(x) -2.25-1.5 0 1 1 2 2 4 3 8 2. Graph the inverse of by making a table. x f(x).25-2.5-1 1 0 2 1 4 2 8 3 Recall: To find an inverse switchx & y values! Unit 7 1
Properties of each graph: y-intercept, x-intercept, asymptote domain,, range,, quadrants,, 2 and2 are inversesof each other! They are reflections over the line, andall andvalues switch! Exponential Functions & Their Inverse of the exponential function : We need to keep going. How do we solve for? We have yet to solve an equation where is located in the exponent! Unit 7 2
To describe 2 in words: is the exponent to the base 2 needed to obtain. We can re-write this sentence using the word logarithm, since logarithm means exponent. is the logarithm to the base 2 of. To abbreviate and write in symbolic form: is the logarithm to the base of. log Inverse Function of the Exponential Function log These two equations are equivalent if 0and 0, (1). Remember: A logarithm IS an exponent. : exponent : base : answer Important! You mustbe able to go back and forth between equivalent logarithmic & exponential forms! Unit 7 3
Ex 1: Write the equation of of 6. 1. Switch and. 2. Solve for. 6 log Ex 2: Write the equation of of log. 1. Switch and. 2. Solve for. log 9 Ex 3: Solve each equation for in terms of. Important! These equations are equivalent. (a) 10 log log (b) 4 1 4 Note: A logarithm of base 10 is called a common logarithm. log Unit 7 4
Transformations of Exponential and Logarithmic Functions Using HSRV Transformations of Functions Recall: Combinations of Transformations: A function involving more than one transformation can be graphed by performing transformations in the following order: 1. Horizontal Shifting 2.Stretching or Shrinking 3. Reflecting 4. Vertical Shifting Unit 7 5
Exponential & Logarithmic Functions H:graph moves right if is positive ( looks negative ) graph moves left if is negative ( looks positive ) S:Vertical Stretch or Shrink and Horizontal Stretch or Shrink (details on next slide) R:Reflection over the axis if is negated Reflection over the axis if is negated V:graph moves up if is positive ( is what it is ) graph moves down if is negative ( is what it is ) Exponential & Logarithmic Functions: Stretch & Shrink S If 1, multiply each -coordinate of by, vertically stretching the graph of by the factor of. If 01, multiply each -coordinate of by, vertically shrinking the graph of by the factor of. If 1, divide each -coordinate of by, horizontally shrinking the graph of by the factor of. If 01, divide each -coordinate of by, horizontally stretching the graph of by the factor of. Unit 7 6
Ex 1: Use the graph of 2 and transformations to sketch the graph of 2 5. Also, find the range & horizontal asymptote of. Hleft 2 units Snone Rreflection over axis Vup 5 units Range:, HA @ Ex 2: Use the graph of log and transformations to sketch the graph of 2log41. Also, find the domain & vertical asymptote of. Need the inverse of the parent function first! Hleft 4 units Svertical stretch by factor (multiply -values by ) Rnone Vdown 1 unit Domain:, VA @ 1 2 Unit 7 7
Ex 3: Use your calculator! Ex 4: You canuse the calculator! Log on Calculator: 1. y= 2. Alpha 3. window 4. Choose #5 logbase Unit 7 8
Ex 5: Determine the domain and vertical asymptote of the graph of log 216 1. Recall: Domain of a logarithmic function is all real numbers greater than zero. Domain:, VA @ Ex 6: Determine the domain and vertical asymptote of the graph of log 35. 3 Domain:, VA @ Unit 7 9
Recall: log These two equations are equivalent if 0and 0, (1). Remember: A logarithm IS an exponent. : exponent : base Important! : answer You mustbe able to go back and forth between equivalent logarithmic & exponential forms! Exponential & Logarithmic Functions: Base 10 and Base e Common Base When an exponential or logarithmic function uses base 10, we call this the common base. log log 10 Equivalent Evaluate: 1. log1 2. log10 3. log100 4. log1000 Unit 7 10
Exponential & Logarithmic Functions: Base 10 and Base e Natural Base When an exponential or logarithmic function uses base e, we call this the natural base. EULER S NUMBER e Given the expression 1 and as, the expression approaches 2.71828. 1. Like, is irrational. 2. 2.71828 3. Used in Exponential Modeling Exponential & Logarithmic Functions: Base 10 and Base e Find an equation for the inverse of. Switch x and y: Solve for y: log Just like a common base has unique notation, so do natural bases: ln Equivalent lnlog Unit 7 11
Exponential & Logarithmic Functions: Base 10 and Base e THE NATURAL LOGARITHM The inverse of is log which is equivalent to ln. Properties of the Natural Logarithm: Domain: 0, Range:, x-intercept: 1, 0 Vertical Asymptote: 0 Ex 1: Write each logarithmic equation in the equivalent exponential form. 1 (a) ln10 (c) log 16 4 1 2 1 16 (b) log 814 3 81 (d) log 6 1 2 36 6 Unit 7 12
Ex 2: Write each exponential equation in the equivalent logarithmic form. (a) 5 625 (c) 10 1 100 log 6254 log 1 100 2 (b) 10 1,000 (d) 9 3 log 1,0003 log1,0003 log 3 1 2 Ex 3: Evaluate each expression. 1 (a) log 64 (b) log 8 1. Set expression equal to a variable or. 1 log 646 log 8 3 2. Re-write logarithmic equation in equivalent exponential form. 2 2 64 1 8 3. Solve exponential equation. (Recall: Need common bases!) 2 2 2 2 6 3 Unit 7 13
Ex 4: Evaluate each expression. (a) log27 (b) 2log 81 log 1 3 27 27 3 3 3 3 log 81 3 81 3 3 4 2 4 8 Ex 5: x-value when y=0 log 4 f : The x-value for f(x) is larger. : Unit 7 14
Ex 6: Multiple Choice: Which of the following is closest to the y-intercept of the function whose equation is 10? (1) 10 (2) 18 (3) 27 (4) 52 y-value when x=0 Recall:... Ex 7: Multiple Choice: On the grid below, the solid curve represents. Which of the following exponential functions could describe the dashed curve? Explain your choice. Dashed graph is increasing Dashed graph is steeper than. Base of dashed graph must be greater than 2.7 Unit 7 15