Evaluate the exponential function at the specified value of x. 1) y = 4x, x = 3. 2) y = 2x, x = -3. 3) y = 243x, x = ) y = 16x, x = -0.

Transcription

1 MAT C TEST 4 REVIEW (CHAP 13) NAME Evaluate the exponential function at the specified value of x. 1) y = 4x, x = 3 2) y = 2x, x = -3 3) y = 243x, x = 0.2 4) y = 16x, x = Solve. 5) The number of bacteria growing in an incubation culture increases with time according to B = 8600(3)x, where x is time in days. Find the number of bacteria when x = 0 and x = 5. 6) If an amount of P dollars is invested at an annual interest rate of r (expressed as a decimal) compounded continuously, the value of the investment after t years is given by V = P ert. Find V if $12,500 is invested at 6.5% for 8 years. Round to the nearest dollar. Convert the equation to logarithmic form. 7) 63 = 216 8) 81/3 = 2 9) 4-2 =

2 Convert the equation to exponential form. 10) log 4 64 = 3 11) log = -2 12) log 27 3 = 1/3 Find the exact value of the expression by first rewriting in exponential form. 13) log232 14) log ) ln e0.6 16) log ) log

3 18) ln 1 e 19) log71 20) log Solve the equation by converting to exponential form. 21) log 3 x = 2 22) log 1/3 x = -4 23) log x 16 = 2 24) log x 1 27 = ) log 3 32 = x 3

4 26) log 6 6 = x 27) log = x 28) log4(6x - 3) = 1 29) log381 = P + 6 Solve. 30) An earthquake was recorded with an intensity which was 316,228 times more powerful than a reference level earthquake, or 316,228 Io. What is the magnitude of this earthquake on the Richter scale (rounded to the nearest tenth)? R = log 10 (I/Io). 31) The population growth of an animal species is described by F(t) = log 3 (2t + 1) where t is measured in months. Find the population of this species in an area 4 month(s) after the species is introduced. Round answer to two significant digits. Write the equation of the inverse of the given function. 32) y = 3 x 33) y = log5 x 4

5 Create a table of values and then graph the inverse functions on the same grid. Use the graphs to complete the following parts. 34) y = 2x and y = log2 x State the following for y = 2x Domain: Range: Equation of asymptote: x-intercept: y-intercept: State the following for y = log2 x Domain: Range: Equation of asymptote: x-intercept: y-intercept: Determine the logarithm in base 10 of the given number to four decimal places. 35) 66 36) ) 3133 Determine the antilog in base 10 of the given number. Round to four decimal places. 38) ) )

6 Find the indicated value. 41) Find f if log f = -6.7, where f is the capacitance of a certain capacitor in farads (F). Give answer to 3 significant digits. 42) Find n if log n = 19, where n is the number of sodium ions in a typical mammalian muscle cell. Solve each problem. 43) The hydrogen potential, ph, of a substance is defined by ph = -log [H+], where [H+] is measured in moles per liter. Find the ph of a sample of lake water whose [H+] is 3.05 x 10-9 moles per liter. (Round to nearest tenth.) 44) The population growth of an animal species is described by F(t) = 400 log (2t + 3) where t is measured in months. Find the population of this species in an area 6 months after the species is introduced. Give answer to two significant digits. Determine the natural logarithm of the number. Round to four decimal places. 45) ) ) 37,900,000 Determine the antilog, base e, of the given number. Round to four decimal places. 48) )

7 Solve the problem. 50) If interest is compounded continuously (daily compounded interest closely approximates this), with an interest rate r (expressed as a decimal), a bank account will double in t years according to r = (ln 2)/t. Find r if the account is to double in 5 years. Round to nearest tenth of a percent. 51) The intensity I of light decreases from its value I0 as it passes a distance x through a medium. Given that x = k ln I0 - ln I, where k is a constant depending on the medium, find x for I = I0 and k = 2.50 cm. Give answer to three significant digits. Use the change of base formula to dtermine the value of the logarithm. Round to the nearest hundredth. 52) log ) log ) log Write as a sum or difference of logarithms. Your results should not contain any exponents or radicals. (EXPAND) 55) log 2 xy 56) log ) log 4 x 6y5 z 5 w6 7

8 58) log 5 25x 7y8 z8 Rewrite the expression as the logarithm of a single quantity. (CONDENSE) 59) log3 q + log3 r 60) log log 3 x 61) 1 log s + 5 log q 2 62) 2 5 log ax log a y 63) logb x5-2logb x 8

9 Solve the exponential equation by writing both sides using the same base. Give exact answers. 1 x 64) = ) 5(9-3x) = ) 4(7-3x) = 1 16 Solve the exponential equation using logarithms. Round answers to three decimal places. 67) 5x = 18 68) ex = 20 9

10 69) e-x = 75 70) e2x + 3ex - 4 = 0 71) 7x + 2 = 3x 10

11 Solve the logarithmic equation. Round answer to three decimal places, where appropriate. 72) log (x + 9) = 1 - log x 73) ln (3x - 5) = ln 10 - ln (x - 2) 74) log 9 (x - 2) + log 9 (x - 2) = 1 75) 3log27x = 2 76) 7ln x = 4 77) 8ln (x-3) = 5 11

12 Solve. 78) The population of a certain country with an annual growth rate of 1.6% is given by the formula P(t) = P t where P0 is the current population. How long will it take for the population to double? Round to the nearest year. 79) There are currently 55 million cars in a certain country, decreasing by 5.5% annually. The number of cars can be determined by the formula N(t) = t where N is measured in millions of cars. How many years will it take for this country to have 32 million cars? Round to the nearest year. 80) If an earthquake measured 7.7 on the Richter scale, what was the approximate intensity of the earthquake? Give answer to two significant digits. R = log 10 (I/Io). 81) If an earthquake measured 5.8 on the Richter scale, what was the intensity of the earthquake? Give answer to two significant digits. R = log 10 (I/Io). 12

13 Determine the type of graph paper on which the graph of the given function is a straight line. 82) y = 5x A) Semilog B) Regular rectangular C) Log-log 83) y = 6x2 A) Regular rectangular B) Log-log C) Semilog 13