FLC Ch 9. Ex 2 Graph each function. Label at least 3 points and include any pertinent information (e.g. asymptotes). a) (# 14) b) (# 18) c) (# 24)

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1 Math 5 Trigonometry Sec 9.: Exponential Functions Properties of Exponents a = b > 0, b the following statements are true: b x is a unique real number for all real numbers x f(x) = b x is a function with domain (, ) b m = b n iff m = n since f(x) = b x is a - function If b > and m < n, then b m < b n. That is, f(x) = b x with b > is an increasing function. If 0 < b < and m < n, then b m > b n. That is, f(x) = b x with 0 < b < is a decreasing function. Defn If b > 0 and b, then f(x) = b x is the exponential function with base b. Characteristics of the Graph of f(x) = b x ) The points (, b ), (0,), and (, b) are on the graph. ) If b >, then f is an increasing function; if 0 < b <, then f if a decreasing function. ) The x-axis is a horizontal asymptote. 4) The domain is (, ) and the range is (0, ). Ex (# 4, #0) If f(x) = x and g(x) = ( 4 )x, find f( ) and g ( ). Ex Graph each function. Label at least points and include any pertinent information (e.g. asymptotes). a) (# 4) b) (# 8) c) (# 4) f(x) = 4 x f(x) = ( ) x f(x) = x Page of 0

2 Ex Sketch the graph of f(x) = ( )x. Then refer to it and using graphing techniques to graph each function. a) (# 0) b) (# ) f(x) = ( x ) + 4 f(x) = ( x 4 ) Ex 4 Solve each equation. a) (# 48) b) (# 54) 5 p+ = 5 ( ) x 6 = 8 x+ c) (# 58) d) ( 5) x = ( x+ 5 ) x 4x = 7 Compound Interest If P dollars are deposited in an account paying an annual rate of interest of r compounded (paid) n times per year, then after t years the account will contain A dollars, where A = P ( + r n ) nt or FV = PV ( + r n ) nt Continuous Compounding If P dollars are deposited at a rate of interest r compounded continuously for t years, the compound amount in dollars on deposit is A = Pe rt. Page of 0

3 Ex 5 (# 64) PP Find the future value and interest earned if $56,780 is invested at 5.% compounded a) quarterly for quarters b) continuously for 5 years Ex 6 Find the required annual interest rate to the nearest tenth of a percent to double your money if interest is compounded quarterly for 8 years. What if it s compounded continuously? Sec 9.: Logarithmic Functions Ex Is the inverse of f(x) = b x a function? If so, find it. Logarithm: Meaning of y = log b x y R and b, x R + where b, y = log b x iff x = b y. Logarithmic Function If b > 0, b, andx > 0, then f(x) = log b x defines the logarithmic function with base b. Page of 0

4 Characteristics of the Graph of f(x) = log b x ) The points ( b, ), (, 0), and (b, ) are on the graph. ) Ifb >, then f is an increasing function; if0 < b <, then f is a decreasing function. ) The y-axis is a vertical asymptote. 4) The domain is (0, ) and the range is(, ). Properties of Logs For x, y, b > 0, b, and any real number r: log b xy = log b x + log b y log b x y = log b x log b y log b x r = r log b x Theorem on Inverses For b > 0, b, log b b x = x for b log b x = x for Ex 7 Write an equivalent statement in log form for each statement. a) (# 4) b) (# 6) c) 5 = 0 4 = e /5 5 = e Ex 8 Write an equivalent statement in exp form for each statement. a) (# 8) b) (# 0) c) log 5 5 = log 4 = ln x = Ex 9 Solve each log equation. a) (# 6) b) (# 0) c) (# 8) x = log 6 6 x = log 5 4 x = log 5 5 Ex 0 (# 4 ) Sketch the graph of f(x) = log x then use graphing techniques to graph g(x) = log (x + ). Find its domain and range. Next, find the domain and range of g(x 0) +. Page 4 of 0

5 Ex (# 8 ) Sketch the graph of f(x) = log / x then use graphing techniques to graph g(x) = log / (x ). Find its domain and range. Ex Use properties of logs to rewrite each expression. Simplify if possible and assume all variables represent positive numbers. a) (# 56) b) (# 60) log 5 log p m5 n 4 t Ex Write each expression as a single logarithm with coefficient. Assume all variables represent positive real numbers a) (# 64) log y p q log y p q b) (# 68) log 6 p log 8p 4 Ex 4 (# 74) Given log and log , find log 6 0. Page 5 of 0

6 Sec 9.: Evaluating Logarithms: Equations and Applications The two most common bases for logarithms are 0 and e. Common Logarithm For all positive numbers x, Natural Logarithm For all positive numbers x, Change-of-Base log x log0 x. ln x log e x. For any positive real numbers x, a, and b, where a and b, log a x log b x. log b a Property of Logarithms If x, y, b 0, b, then x y iff log x log y. 4 Ex 5 (# ) Use a calculator to find an approximation to 4 decimal places of ln b b e. 9 Ex 6 (# 4) Find the ph of crackers with hydronium ion concentration of.9 0. Use ph log H O, where H O is the hydronium ion concentration (in moles per liter). Ex 7 (# 40) Use change-of-base and a calculator to find log 9 5 to 4 decimal places. Ex 8 (# 46) Given x log x f, evaluate a) f b) f log log c) f Page 6 of 0

7 Ex 9 (# 50) The magnitude of an earthquake, measured on the Richter scale, is log0 I I, where I is the amplitude registered on a seismograph 00 km from the epicenter of the earthquake, and I 0 is the amplitude of an earthquake of a certain (small) size. On June 6, 999, the city of Puebla in central Mexico was shaken by an earthquake that measured 6.7 on the Richter scale. Express this reading in terms of I 0. 0 Ex 0 Evaluate. a) log 0.00 = b) log 6 = c) log = d) log ( ) = e) ln e = f) log = g) log ( 7 ) = h) log 5 5 ( ) = i) log ( 4) = Ex Solve. a) b) 8 x 7 = 6 x 5 x+ = 0 x c) d) 8 0 x 7 = log(x ) + log(x + ) = log Page 7 of 0

8 e) f) ln(5x ) + ln(7x ) = ln(5x + 7) log x+7 (6) log x+7 4 = g) h) [Hint: Multiply each side by e x.] [sinhx = ] 4 x + x+ = 0 e x e x = Practice Problems ) Graph the following exponential function. y x4 ) Solve the following exponential equations algebraically. Show all your work and give an exact answer. a. x 9 4 b. x 4 x 6 8 c. x 4 5 ) Find the balance A for P dollars invested at rate r for t years and compounded n times. a. P = $000, r = 6.5%, t = 0 years, compounded quarterly. b. P = $000, r = 7%, t = 8 years, compounded continuously. 4) Find the required annual interest rate to the nearest tenth of a percent for $48,000 to grow to $78,86.94 when compounded semiannually for 5 yr. 5) Solve the following logarithmic equations algebraically. Show all your work and give an exact answer. a. log x b. log x 5 c. log 4 x 6 6 6) Graph the following logarithmic function. y = x 4 log Page 8 of 0

9 7) Use the properties of logarithms to expand or condense the following log expressions. a. y x log b b. log log y log z 4 a x a a z 8) Find the ph for lye soap which has a hydronium ion concentration of. 0 moles per liter. 9) Find the O H for drinking water which has a ph of ) Use the change of base theorem to find an approximation for log 5. ) Given f(x) = x, evaluate the following. a. f (log 7) b. f [log (ln )] c. f [log ( ln )] 4 ) a. Find the decibel rating of a sound having an intensity of 00,000 I 0. b. If the intensity of the sound is doubled, by how much is the decibel rating increased? ) a. Find the Richter scale rating for an earthquake having an amplitude of,000,000 I 0. b. Express the magnitude of a 6.7 earthquake (on the Richter scale) in terms of I 0. 4) Solve. x a. 4 6 b. x5 x 4x c. 6e 4t.06 d ) a. log (x ) + log 0x = log 0 b. ln x + ln (x ) = c. 4x log x log log6 6 6 d. log (x + 9) = + log (x + ) 6) Elena McDuff wants to buy an $8,000 painting. She has saved $6,000. Find the number of years (to the nearest tenth) it will take for her savings to grow to $8,000 at 5.7% compounded monthly. 7) At what interest rate will $,500 grow to $4,67.04 if invested for 0 years and interest is compounded quarterly? Answers. HA: y = key point : (4, 0). a. x = b. x = c. x = 65. a. $8. b. $ % x y Page 9 of 0

10 5. a. x = 4 b. x = 4 c. x y 4 6. VA: x = ; exponential form of the equation: x x y a. logb y logbx 4 logbz b log a x yz a. 7 b. ln.0986 c. ln = ln = ln a. 50 b. about decibels. a. 6 b. about 5,000,000 I 0 4. a..764 b c d a. b..69 c. 4.5 d. Ø years 7. 6.% Page 0 of 0