ALGEBRA II

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

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 =

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

Now, let’s play some If log = 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

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) x = b) 10 -x = d) x = 8.53

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

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

Solve for x. Give exact answers. Examples. log 3 x = = 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.

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

A practical use of logarithms is the Richter Scale, used to measure the magnitude of an earthquake.

PROPERTIES OF LOGARITHMS x 4 x 7 = x 4+7 = x 11 x 20 x 10 = x = 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

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

9) Solve for x. a) 5 x = 94 b) 2 x = c) 60.4x = 121 d) x = ) If log2 = and log3 = 0.477, find log48.

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

Now, just for fun, let’s make the Gizmoraptor go backwards. Of course, the equation must change. Q 5 R ) 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 ?

*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?

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))

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

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

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

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

Exponential Variation – Fractal Flames