1. ALGEBRA II HONORS @

2. EVERYTHING YOU WANTED TO KNOW ABOUT LOGARITHMS, BUT DID NOT KNOW WHAT TO ASK.

3. 1) Find x in 10 x = 59 to 2 decimal places. 2) Now press log59 on your calculator. What happens? 3) Solve. Round your answers to 4 decimal places. a) 10 x = 3970 b) 10 x = 0.001492

4. LOGARITHM (LOG): means logical arithmetic. A COMMON LOG is a logarithm of base 10 (the kind that your calculator will do). If 7 2 = 49, then log 7 49 = 2 If 4 3 = 64, then log 4 64 = 3 If, then log 36 6 = If b x = a, then…..log b a = x

5. Now, let’s play some If log 7 343 = 3, then 7 3 = 343 If then,oops, can’t do it! Why? By definition, the base and the argument must be positive. If, then

6. Therefore, a LOGARITHM is an EXPONENT. If b x = a, then log b a = x with a, b > 0 and b 1 a is called the argument, b is the base, and x is the exponent. 4)Solve. Round your answers to 4 decimal places when necessary. a)10 x = 457c) 2.3 10 x = 0.1776 b) 10 -x = 0.0247d) 3.5 10 2x = 8.53

7. If a logarithm is not in base 10, we change use the change of base formula to transform it. 2 x = 3 log2 x = log3 Take the log of both sides. xlog2 = log3 (you will understand why very soon) x = log3 log2 x 1.5850

8. 5)Solve for x. Round your answers to 4 decimal places when necessary. a)5 x = 6.2 b) 4.13 -x = 987 c) 4 23 2x = 371

9. Solve for x. Give exact answers. Examples. log 3 x = -4 3 -4 = x = x log 2 8 = x 2 x = 8 x = 3 When the variable is the base, solve by raising both sides to the reciprocal of the exponent of the variable.

10. 6) Find x. a) log 5 x = 5 b) log 12 x = -4 c) log 4 16384 = x d) e) log -4 16 = x f) g) h) i) log x 32 = -5

11. A practical use of logarithms is the Richter Scale, used to measure the magnitude of an earthquake. www.seismo.unr.edu/ftp/pub/louie/class/100/magnitude.html

12. PROPERTIES OF LOGARITHMS x 4 x 7 = x 4+7 = x 11 x 20 x 10 = x 20+10 = x 30 log(5 6) = log5 + log6 log(21 12) = log21 + log12 PROPERTIES 1) log(xy) = logx + logy 2) (x 4 ) 7 = x 47 = x 28 log(7 5 ) = 5log7 3) logx y = ylogx

13. 7) Express as a single log with a single argument. a) log 6 5 + 2log 6 4 b) log 3 4 – 5log 3 2 c) d) log 3 30 + log 3 49 – log 3 42 8) Solve for x.log 2 x 3 – log 2 27 = 3

14. 9) Solve for x. a) 5 x = 94 b) 2 x = 0.149 c) 60.4x = 121 d) 3.6 -2.5x = 1066 10) If log2 = 0.301 and log3 = 0.477, find log48.

15. INVERSES OF FUNCTIONS We have a computer called the Gizmoraptor that does computations for us. R=5Q + 2 Q 5 R 27 -3 -13 3 0 2 11) What are the ordered pairs we just made? (5, 27), (-3, -13), (, 3), and (0, 2)

16. Now, just for fun, let’s make the Gizmoraptor go backwards. Of course, the equation must change. Q 5 R 27 -3 -13 3 0 2 11) What are the ordered pairs we just made? (27, 5), (-13, -3), (3, ), and (2, 0) 12) What do you notice about the ordered pairs from the two functions R = 5Q + 2 and ?

17. *Get a sheet of graph paper and draw a Cartesian Coordinate System. *Draw a figure in the 2 nd quadrant and note a few of its ordered pairs. *Draw the line y = x on your coordinate system. We note this line has special properties. *Fold your coordinate system using the line y = x as the seam of the fold. *Trace your figure, then draw it on your coordinate system. *What do you notice about the ordered pairs of the second figure you drew?

18. Recall the equations from before : R = 5Q + 2 and 13)a) Find f(g(2)) b) Find g(f(2)) c) Find f(g(-3)) d) Find g(f(-3)) e) Find f(g(x)) f) Find g(f(x))

19. PROPERTIES OF INVERSES 1)The ordered pairs are reversed. 2)The graphs are reflected about the line y = x. 3) f(g(x)) = g(f(x)) = x For additional information, access http://www.purplemath.com/modules/invrsfcn.htm

20. Find each inverse. f(x) = 5x + 2original problem y = 5x + 2let y be f(x) x = 5y + 2switch the x’s and the y’s solve for y let f -1 (x) be y

21. 14) Find each inverse. a) f(x) = 3x - 7 b) g(x) = 0.2x + 3 c) h(x) = x 2 d) m(x) = log 2 x

22. MATHEMATICAL MODELS Exponential variation : y = a b x where x = time, and y = the amount of something Go over Problems 3 & 4 on pages 436- 437.

23. Exponential Variation – Fractal Flames http://draves.org/pix/2004-12-20-exponential/gen/Index-001.jpg