"The Relationship" Logarithms are the "opposite" of exponentials, just as subtraction is the opposite of addition and division is the opposite of multiplication. Logs "undo" exponentials. Technically speaking, logs are the inverses of exponentials. In practical terms, | have found it useful to think of logs in terms of The Relationship: The Relationship y=b is equivalent to | logy(y) =x| (means the exact same thing as) On the left-hand side above is the exponential statement "y = b*". On the right-hand side above, "log,()" is the equivalent logarithmic statement which is pronounced as \"log-base-b of 3"; "b" is called "the base of the logarithm", just as 1, is the base in the exponential expression \"b*'. And, just as the base 1, in an exponential is always positive and not equal to 1, so also the base 1, fora logarithm is always positive and not equal to 1. Whatever is inside the logarithm is called the "argument" of the log. Note that the base in both the exponential equation and the log equation (above) is "b", but that the x and y switch sides when you switch between the two equations. log,(x)=y If you can remember this (that whatever had been the argument of the log becomes the "equals" and whatever had been the "equals"becomes the exponent in the exponential, and vice versa), then you shouldn't have too much trouble with logarithms. (The term "The Relationship" comes from the websitewww.purplemath.com. You will not find it in your text, and your teachers and tutors will have no idea what you're talking about if you mention it to them. "The Relationship" is entirely non-standard terminology.) By the way: If you noticed that the variables are switched between the two boxes displaying "The Relationship", you've got a sharp eye. | did that on purpose, to stress that the point is not the variables themselves, but how they move. Graphing an Exponential To graph an exponential, you need to plot a few points, and then connect the dots and draw the graph, using what you know of exponential behavior. Here is an example: Graphy=3* Since 3* grows so quickly, | will not be able to find many reasonably-graphable points on the right-hand side of the graph. And 3* will get very small very quickly on the left-hand side of the graph, so | probably won't find many useful plot-points there, either. So | will find a few plot-points in the middle, close to the origin, and then draw the graph from there. L3]373 117 % 0.04 m X0, y >0 xm, y e ve=x' o I -~ Parent Function Graph Parent Function Graph y - x =[x Linear, Odd Absolute Value, Even Domain: -00,00 Domain: (-20,00) Range: (-00,00 Range: [0,0) End Behavior: x - 0, y End Behavior: x- -30,y- x-0. yo y =x y - vx Quadratic, Even Radical, Neither Domain: (-20,00) Domain: [0,20) Range: [0,) Range: [0,0) End Behavior: x - -20, y-+30 End Behavior: x-+ 00, y - y-x y - NX Cubic, Odd Cube Root, Odd Domain: -00, 00 Domain: (-00,00) Range: (-00,00) Range: (-00,00) End Behavior: End Behavior: x- -, y+ x- -50. y + x-0, y-0 y-b", b>1 y = log.(x), b>1 Exponential, Neither Log, Neither Domain: (-60,00) Domain: (0,00 Range: (0,00) Range: (-00,00) End Behavior: End Behavior: x- -0,y-0 x20'.y--2 x -+co, y -0 x-0, y- y - Rational (Inverse), Odd Rational (Inverse Squared), Even Domain: (->,0)(0,20) Range: (-x,0)(0.) Domain: (-20,0) (0,0) Range: (0,30 ) End Behavior: x- -4,y+0 End Behavior: x-0, y+0 x - -00, y=0 y =int(x)=[x] y - C Greatest Integer, (y = 2 in the graph) Neither Constant, Even Domain: (-00,00) Domain: (-00,00) Range: (y:ye Z) (integers) Range: (y:y =c} End Behavior: End Behavior: x-+00, y -cFor questions 1-5, choose the graph that goes with the given function. 1y=3 \f\f\f3. y = log3X A. 3 2 O - NW B. O C. 2 O D. . O E. K 3 2 -O E. K -2 Not O N F. K 4 -24. y = log3(1 - x) O N A. O B. O C. O D. O E. KO E. H O -2\fO E. O N F. -2For questions 6-10, choose the function that goes with the given graph. 6. -3 -2 2 O A. y = log4(-X) O B.y=-log2(x) O C. y = 5x D. y = ( ) O E. y = log7(x) O F. y = log2(5)\f\f\f\fFor question 11, choose the correct equation. 11. Which equation below is equivalent to the equation z = d*? O Aw=d? OB.d=2" O C.w=log,(d) O D.d=w O E.w=logy(2) O F.z=logg(w) For questions 12-18, give the domain and range of the function. Use interval notation. (Answers may repeat, so check the available choices carefully.) NOTE: The graph appears to stop on the left, but it continues to drop vertically. O A. Domain: (c0,00) Range: [0,00) O B. Domain: (00,00) Range: (4,00) O C. Domain: (4,00) Range: (0,00) O D. Domain: (4,00) Range (00,00) O E. Domain: [4,00) Range: [0,00) O F. Domain: (,0) Range: (00,00) 13. -3 -2 2 - 2 O A. Domain: (-00, 00) Range: [0,00) O B. Domain: [0, co) Range: (-00, 00) O C. Domain: (4,00) Range: (-00, 00) O D. Domain: [4,00) Range: [4,00) O E. Domain: (-oo, 00) Range: (0,00) O F. Domain: (-00, 00) Range: (4,00)-2 14. -2 -1 - 2 O A. Domain: (-00, 00) Range: (0,00) O B. Domain: [4,00) Range: [4,00) O C. Domain: [0, 00) Range: (-00, 00) O D. Domain: (-oo, 00) Range: (4,00) O E. Domain: (-4,00) Range: (0,00) O F. Domain: [0,co) Range: (-00, 00)O A. Domain: [4,00) Range: [4,00) O B. Domain: (4,00) Range: (4,00) O C. Domain: (4,00) Range: (00,00) O D. Domain: [0,00) Range (00,00) O E. Domain: (-00,00) Range: (4,00) O F. Domain: (0,0) Range: (~00,00) 16. | } } } -8 & O A. Domain: (-00,00) Range: (0,00) O B. Domain: [-4,00) Range: [0,00) O C. Domain: [0,00) Range: (00,00) O D. Domain: (0,00) Range: (o0,00) O E. Domain: (-00,00) Range (4,00) O F. Domain (4,00) Range: (0,00) - -8 O A. Domain: [0,00) Range: (00,00) O B. Domain [0,00) Range (o0,00) O C. Domain: [4,00) Range: [4,00) O D. Domain: (o0,00) Range: (4,00) O E. Domain: [-4,00) Range: [0,00) O F. Domain: (0,00) Range: (c0,00) 1.5 -0.5 18. -2. -1.5 -0.5 0.5 1.5 -0.5 1.5 O A. Domain: (-00, 00) Range: [0,00) O B. Domain: (4,00) Range: (4,00) O C. Domain: [4,00) Range: [4,00) O D. Domain: (4,00) Range: (-00, 00) O E. Domain: (-00,0) Range: (-00, 00) O F. Domain: (-oo,0] Range: (-00, 00)For questions 19 and 20, choose the correct statement. 19. What is ALWAYS true about the graph of the function f(x) = a* (where "a" is any positive integer)? O A. It goes through the point (1,0). O B. It goes through the point (0,1). O C. It goes through the point (1,1). O D. It goes through the origin. O E. It touches both the x-axis and the y-axis.None of the answers are correct. O F. None of the answers are correct 20. What is ALWAYS true about the graph of the function f(x) = log(x)? O A. None of the answers are correct. O B. It goes through the point (0,1). O C. It goes through the point (1,1). O D. It goes through the origin O E. It touches both the x-axis and the y-axis. O F. It goes through the point (1,0)