Exponentials, Logarithms, and Inverses Created by Educational Technology Network. www.edtechnetwork.com 2009

Exponential Functions Logarithm Functions Hodgepodge Inverses (Algebra) Inverses (Graphs) 100 100 100 100 100 200 200 200 200 200 300 300 300 300 300 400 400 400 400 400 500 500 500 500 500

Exponentials – 100 Points QUESTION: (30 sec.) Simplify ANSWER: 7v5/2

Exponentials – 200 Points QUESTION: ANSWER: (1 min) Convert to a logarithm: 36=729 ANSWER: Log3729 = 6

Exponentials – 300 Points QUESTION: ANSWER: (1 min) Simplify the following: 3x/3y 3x•3y (3x)y ANSWER: 3x-y 3x+y 3x•y

Exponentials – 400 Points QUESTION: ANSWER: (2 min) Solve: 5(67x)=13

Exponentials – 500 Points QUESTION: ANSWER: (4 min) Solve: e2x – 6ex +8 = 0 (Hint: factor) ANSWER: ex = 2  x = 0.69315 ex = 4  x = 1.38629

Logarithms– 100 Points QUESTION: ANSWER: (2 min) Expand the following: Log2 √(2x(x2+2)) ANSWER: ½ • [Log22+Log2x+Log2(x2+2)]

Logarithms – 200 Points QUESTION: ANSWER: (2 min) Write the expression as a single logarithm 3 ln 2 – ½ ln x + ln (x+1) ANSWER: ln [ (8•(x+1) ) / ( √(x) ) ]

Logarithms – 300 Points QUESTION: ANSWER: List the Product, Quotient and Power of a Power Logarithm properties. (1 min) ANSWER: Product: Logb(mn) = Logbm + Logbn Quotient: Logb(m/n) = Logbm – Logbn Power: Logbmn = n•Logbm

Logarithms – 400 Points QUESTION: ANSWER: (1 min)What is the change of base formula ANSWER: Logba = Log (a) / Log (b)

Logarithms – 500 Points QUESTION: ANSWER: (4 minutes) Solve for x: log3 (x+9) = log3((x-3)/(x+2)) x+9=(x-3)/(x+2) (x+9)(x+2) = x-3 x2+11x+18= x-3 x2+10x+21=0 (x+3) (x+7) = 0 x = -3, -7

Hodgepodge – 100 Points QUESTION: ANSWER: (2 min) Expand: Log ( (7xy5) / (√(x-2) ) ) ANSWER: Log 7 + Log x + 5•Log y + ½ Log (x-2)

Hodgepodge – 200 Points QUESTION: ANSWER: (1 min) What is the base of our general Log? What about ln? ANSWER: Log is base 10 unless specified Ln is the natural logarithm with a base of e

Hodgepodge – 300 Points QUESTION: ANSWER: Explain using properties why taking the natural log of e7x gives you 7x ANSWER: Ln (e7x) = 7x •Log (e) = 7x • 1 = 7x

Hodgepodge – 400 Points QUESTION: ANSWER: What mathematician was born on Pi Day? ANSWER: Albert Einstein

Hodgepodge – 500 Points QUESTION: ANSWER: Simplify and solve x = ±1 does -1 make sense?

Inverses (Algebra) – 100 Points QUESTION: Find the inverse of f(x)=3x+2. ANSWER: f -1(x)=(x-2)/3

Inverses (Algebra) – 200 Points QUESTION: If regular gasoline is selling for $3.95 per gallon, the price of any particular purchase p is a function of the number of gallons g in that purchase. Write this as a function and find its inverse ANSWER: Function: p=3.95•g Inverse: g=p/3.95

Inverses (Algebra) – 300 Points QUESTION: ANSWER: y=(7/x)-4

Inverses (Algebra) – 400 Points QUESTION: Write out the steps for either of the two algebra styles we have learned to determine the inverse of a function ANSWER: Answers may vary

Inverses (Algebra) – 500 Points QUESTION: How could you use composition of functions to check that f(x) and g(x) are inverses of each other? ANSWER: Test f(g(x)) and/or g(f(x)) and check that they equal x.

Inverses (Graph) – 100 Points QUESTION: What is the line of symmetry for functions and their inverse? ANSWER: y=x

Inverses (Graph) – 200 Points QUESTION: Graph the following function then graph its inverse: y=3x+1 ANSWER: Inverse equation: y=(x-1)/3

Inverses (Graph) – 300 Points QUESTION: Given the following functions, which have inverses that are functions? If it does not have one explain why. a) b) c) ANSWER: a & b have inverse functions, c does not because it fails the “horizontal line test”

Inverses (Graph) – 400 Points QUESTION: How do the domain and range of a function relate to the domain and range of its inverse ANSWER: They are switched, why?

Inverses (Graph) – 500 Points QUESTION: How could I use a graph to check that my functions are inverses of each other? ANSWER: y=x is the line of symmetry, this is made clear when the coordinates flip (x,y)  (y,x)