52. Error Analysis A student has broken the natural logarithm key on his calculator, so he decides to use the Change of Base Formula to find ln 100. Explain his error and find the correct answer.
53. Satellite The battery power available to run a satellite is given by the formula P = 50 e − t 250 , p equals 50 . e super negative , t over 250 end super . comma  where P is power in watts and t is time in days. For how many days can the satellite run if it requires 15 watts of power?

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Determine whether each statement is always, sometimes, or never true.

54. l n e x ≥ 1 l n , e to the x , greater than or equal to 1
55. l n e x = ln e x + 1 l n , e to the x , equals ln , e to the x , plus 1
56. l n t = log e t l n t equals , log base e , t
57. Space Use the formula for maximum velocity v = − 0 . 0098 t + c ln R . v equals negative 0 . , 0098 to the t , plus c ln r .  Find the mass ratio of a rocket with an exhaust velocity of 3.1 km/s, a firing time of 50 s, and a maximum shuttle velocity of 6.9 km/s.

Biology The formula H = 1 r ( l n P − l n A ) h equals , 1 over r , open l n p minus l n eh close  models the number of hours it takes a bacteria culture to decline, where H is the number of hours, r is the rate of decline, P is the initial bacteria population, and A is the reduced bacteria population.

58. A scientist determines that an antibiotic reduces a population of 20,000 bacteria to 5000 in 24 hours. Find the rate of decline caused by the antibiotic.
59. A laboratory assistant tests an antibiotic that causes a rate of decline of 0.14. How long should it take for a population of 8000 bacteria to shrink to 500?

C Challenge

Solve each equation.

60. 1 3 l n x + l n 2 − l n 3 = 3 1 third , l n x plus l n 2 minus l n 3 equals 3
61. l n ( x + 2 ) − ln 4 = 3 l n open x plus 2 close minus ln 4 equals 3
62. 2 e x − 2 = e x + 7 2 . e super x minus 2 end super . equals , e to the x , plus 7

63. Error Analysis Consider the solution to the equation ln ( x − 3 ) 2 = 4 open x minus 3 close squared . equals 4  below. In Problem 2 you saw that there are two solutions to this equation, 3 + e 2 3 plus , e squared  and 3 − e 2 . 3 minus , e squared , .  Why do you get only one solution using this method?

    
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64. Technology In 2008, there were about 1.5 billion Internet users. That number is projected to grow to 3.5 billion in 2015.
    1. Let t represent the time, in years, since 2008. Write a function of the form y = a e c t y equals . eh e super c t end super  that models the expected growth in the population of Internet users.
    2. In what year are there 2 billion Internet users?
    3. In what year are there 5 billion Internet users?
    4. Solve your equation for t.
    5. Writing Explain how you can use your equation from part (d) to verify your answers to parts (b) and (c).

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