POD. A) Graph: y = 2e x +2 B) Evaluate: (e 2x e x ) 2 2e -x. e 7x 2
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1 POD A) Graph: y = 2e x +2 B) Evaluate: (e 2x e x ) 2 2e -x e 7x 2
2 4.4 Evaluate Logarithms & Graph Logarithmic Functions What is a logarithm? How do you read it? What relationship exists between logs and exponents? What is the definition? How do you rewrite log equations? What are two special log values? What is a common log? A natural log? What logs can you evaluate using a calculator?
3 Evaluating Log Expressions We know 2 2 = 4 and 2 3 = 8 But for what value of y does 2 y = 6? Because 2 2 <6<2 3 you would expect the answer to be between 2 & 3. To answer this question exactly, mathematicians defined logarithms.
4 Definition of Logarithm to base b Let b & x be positive numbers & b 1. The logarithm of x with base b is denoted by log b x and is defined: log b x = y if b y = x This expression is read log base b of x The function f(x) = log b x is the logarithmic function with base b.
5 The definition tells you that the equations log x = y and b y = x are b equivalent. Rewriting forms: To evaluate log 9 = x ask yourself 3 Self 3 to what power is 9? 3 2 = 9 so log 3 9 = 2
6 Log form Exp. form log 2 16 = = 16 log = 1 log 3 1 = 0 log 10.1 = = = =.1 log = 6
7 Evaluate log 81 = x4 3 x = 81 3 Log = x 3 5 x = 125 Log 256 = 4 x 4 4 x = 256 Log 2 (1/32) = x-5 2 x = (1/32)
8 Evaluating logarithms now you try some! Log 4 16 = 2 Log 5 1 = Log 4 2 = 0 ½ (because 4 1/2 = 2) Log 3 (-1) = undefined (Think of the graph of y=3 x )
9 You should learn the following general forms!!! Log b 1 = 0 because b 0 = 1 Log b b = 1 because b 1 = b Log b b x = x because b x = b x
10 Natural logarithms log e x = ln x e ln means log base e
11 Common logarithms log 10 x = log x Understood base 10 if nothing is there.
12 Common logs and natural logs with a calculator log 10 button ln button **Only common log and natural log bases are on a calculator.
13 Keystrokes Expression Keystrokes Display Check a. log b. ln e
14 Tornadoes The wind speed s (in miles per hour) near the center of a tornado can be modeled by: s = 93 log d + 65 where d is the distance (in miles) that the tornado travels. In 1925, a tornado traveled 220 miles through three states. Estimate the wind speed near the tornado s center.
15 Solution s = 93 log d + 65 = 93 log (2.342) + 65 = Write function. Substitute 220 for d. Use a calculator. Simplify. ANSWER The wind speed near the tornado s center was about 283 miles per hour.
16 What is a logarithm? How do you read it? A logarithm is another way of expressing an exponent. It is read log base b of y. What relationship exists between logs and exponents? What is the definition? log a x = y if a y = x How do you rewrite logs? The base with the exponent on the other side of the =. What are two special log values? Log b 1=0 and log b b=1 What is a common log? A natural log? Common log is base 10. Natural log is base e. What logs can you evaluate using a calculator? Base 10
17 Hw 4.4 Page 255, 3-6, 8-15, even, even, 37,38,42,43, 47, 49, 51 We have covered through 26.don t put this off...there are 27 problems!
18 A) Log 15 1 = POD B) Log 3 3 = C) Log = D) Log 3 27 =
19 4.4 Day 2 How do you use inverse properties with logarithms? How do you graph logs?
20 g(x) = log b x is the inverse of f(x) = b x f(g(x)) = x and g(f(x)) = x Exponential and log functions are inverses and undo each other.
21 So: g(f(x)) = log b b x = x f(g(x)) = b log bx = x 10 log2 = 2 Log 3 9 x = Log 3 (3 2 ) x =Log 3 3 2x =2x 10 logx = x Log x = 3x
22 Use Inverse Properties Simplify the expression. a. 10 log4 b. log 25 x 5 SOLUTION a. 10 log4 = 4 log b x b = x b. log 25 x = 5 log = 5 = 5 log 5 2x 2x (5 2 ) x Express 25 as a power with base 5. Power of a power property log b x = x b
23 Use Inverse Properties Simplify the expression. a. 8 log 8x SOLUTION 8 log 8x = x 11. log 7 3x 7 SOLUTION log 7 3x = 3x 7 log b x b = x logaa x = x Exponent form Log form Log form Exponent form
24 Finding Inverses Find the inverse of: y = log 3 x By definition of logarithm, the inverse is y=3 x OR write it in exponential form and switch the x & y! 3 y = x 3 x = y
25 Finding Inverses cont. Find the inverse of : Y = ln (x +1) X = ln (y + 1) e x = y + 1 Switch the x & y Write in exp form e x 1 = y solve for y
26 Find Inverse Properties Find the inverse of the function. a. y = 6 x b. y = ln (x + 3) SOLUTION a. From the definition of logarithm, the inverse of y = 6 x is y= log x. b. y = ln (x + 3) x = ln (y + 3) e x = (y + 3) e x 3 = y 6 Write original function. Switch x and y. Write in exponential form. Solve for y. ANSWER The inverse of y = ln (x + 3) is y = e x 3.
27 Use Inverse Properties 14. Find the inverse of y = 4 x SOLUTION From the definition of logarithm, the inverse of y = 4 x is y = log x. 15. Find the inverse of SOLUTION y = ln (x 5) x = ln (y 5) e x = (y 5) e x + 5 = y 4 y = ln (x 5). Write original function. Switch x and y. Write in exponential form. Solve for y.
28 4.4 Graphing Logs p. 254
29 Graphs of logs y = log b (x-h)+k Has vertical asymptote x=h The domain is x>h, the range is all reals If b>1, the graph moves up to the right If 0<b<1, the graph moves down to the right
30 Graphing a log function Graph the function. a. y = log x 3 SOLUTION Plot several convenient points, such as (1, 0), (3, 1), and (9, 2). The y-axis is a vertical asymptote. From left to right, draw a curve that starts just to the right of the y-axis and moves up through the plotted points, as shown below.
31 Graph y = log 1/3 (x)-1 Plot (1/3,0), (1,-1) & (3,-2) Vert line x=0 is asy. Connect the dots X=0
32 Graph y =log 5 (x+2) Plot easy points (-1,0) & (3,1) Label the asymptote x= 2 Connect the dots using the asymptote. Domain? Rang? X=-2
33 How do you use inverse properties with logarithms? Exponential and log functions are inverses and undo each other. How do you graph logs? Pick 1, the base number, and a power of the base for x.
34 Hw 4.4 Page 255, 3-6, 8-15, even, even, 37,38,42,43, 47, 49, 51
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