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1 Name: Units do not have to be included Log1 Contest Round 2 Theta Logs and Exponents points each 1 Write in logarithmic form: 2 = Evaluate: log 5 0 log 5 8 (log 2 log 6) Simplify the expression log log 9 Leave your answer as an improper fraction. Solve for x: log 7 (log 2 (log 12 x)) = 0 5 Solve for z: z = log loglog Find the last digit in the sum of points each 7 Evaluate: log 9 (8) log 10 (9) log 11 (10) log 096 (095) 8 Express log(1228) in terms of a and b given that log (07) = a and log (625) = b. Any term that does not contain a or b must be expressed as an integer or fraction. 9 Solve for x: 15 = 8ln(x) Solve for x and express your answer in root form. x xxx... = 5
2 6 points each 11 Solve for x and express your answer in root form. (6 ) 8 = 108 ( 6) ( x) 12 A population of ants follows an increasing, exponential growth model, multiplying by a factor of 15 5 every days. If the initial population was 10, what is the population 8 after days? Note that there are no such things as fractions of an ant when talking about population growth. 1 Solve for x in terms of a. 2 log b x = 2 log b (1 a) + 2 log b (1 + a) log b ( 1 2 a a) 1 Solve for x: 9 7 2x+6 x = 9 x 15 Evaluate: log 81 ( 9 27 )
3 Name: Units do not have to be included Log1 Contest Round 2 Alpha Logs and Exponents points each 1 Write in logarithmic form: 2 = Evaluate: log 5 0 log 5 8 (log 2 log 6) Simplify the expression log log 9 Leave your answer as an improper fraction. Which of the following is smaller, 20 or 6 12? 5 Solve for z: z = log loglog Find the last digit in the sum of points each 7 Evaluate: log 9 (8) log 10 (9) log 11 (10) log 096 (095) 8 Express log(1228) in terms of a and b given that log (07) = a and log (625) = b. Any term that does not contain a or b must be expressed as an integer or fraction. 9 Solve for x: log(x) 1 = log(x 9) 10 Solve for x and express your answer in root form. x xxx... = 5
4 6 points each 11 Solve for x and express your answer in root form. (6 ) 8 = 108 ( 6) ( x) 12 A population of ants follows an increasing, exponential growth model, multiplying by a factor of 15 5 every days. If the initial population was 10, what is the population 8 after days? Note that there are no such things as fractions of an ant when talking about population growth. 1 Solve for x in terms of a. 2 log b x = 2 log b (1 a) + 2 log b (1 + a) log b ( 1 2 a a) 1 Evaluate: log 2 81 log log e e Evaluate: log 81 ( 9 27 )
5 Name: Units do not have to be included Log1 Contest Round 2 Mu Logs and Exponents points each 1 Write in logarithmic form: 2 = Evaluate: log 5 0 log 5 8 (log 2 log 6) Simplify the expression log log 9 Leave your answer as an improper fraction. Which of the following is smaller, 20 or 6 12? 5 Find the y-value of the point on the curve y = ln(x)dx when x =, given that when x = 1, y = 8 6 Find the last digit in the sum of points each 7 Evaluate: log 9 (8) log 10 (9) log 11 (10) log 096 (095) 8 Express log(1228) in terms of a and b given that log (07) = a and log (625) = b. Any term that does not contain a or b must be expressed as an integer or fraction. 9 Solve for x: log(x) 1 = log(x 9) 10 Evaluate: log ( x ln()dx 0 + 1)
6 6 points each 11 Solve for x and express your answer in root form. (6 ) 8 = 108 ( 6) ( x) 12 A population of ants follows an increasing, exponential growth model, multiplying by a factor of 15 5 every days. If the initial population was 10, what is the population 8 after days? Note that there are no such things as fractions of an ant when talking about population growth. 1 Solve for x in terms of a. 2 log b x = 2 log b (1 a) + 2 log b (1 + a) log b ( 1 a a) 2 1 Evaluate: log 2 81 log log e e Given that f(x) = ln e 2lne( x )lne x evaluate the derivative of f(x) at x = 2. That is, find f (2)
7 Name: Units do not have to be included Log1 Contest Round 2 Theta Logs and Exponents Answer Key points each 1 Write in logarithmic form: 2 = 1 8 log 2 ( 1 8 ) = 2 Evaluate: log 5 0 log (log 2 log 6) Simplify the expression log log 9 Leave your answer as an improper fraction. 7 2 Solve for x: log 7 (log 2 (log 12 x)) = Solve for z: z = log loglog points each 6 Find the last digit in the sum of Evaluate: log 9 (8) log 10 (9) log 11 (10) log 096 (095) 1 8 Express log(1228) in terms of a and b given that log (07) = a and log (625) = b. Any term that does not contain a or b must be expressed as an integer or fraction. a b Solve for x: 15 = 8ln(x) e11 10 Solve for x and express your answer in root form. 5 5 x xxx... = 5
8 6 points each 11 Solve for x and express your answer in root form. (6 ) 8 = 108 ( 6) ( x) 12 A population of ants follows an increasing, exponential growth model, multiplying by 90 a factor of 15 5 every days. If the initial population was 10, what is the population 8 after days? Note that there are no such things as fractions of an ant when talking about population growth. 1 Solve for x in terms of a. 2 log b x = 2 log b (1 a) + 2 log b (1 + a) log b ( 1 2 a a) 1 Solve for x: 9 7 2x+6 x = 9 x x = a 1 15 Evaluate: log 81 ( 9 27 ) 1 8
9 Name: Units do not have to be included Log1 Contest Round 2 Alpha Logs and Exponents Answer Key points each 1 Write in logarithmic form: 2 = 1 8 log 2 ( 1 8 ) = 2 Evaluate: log 5 0- log (log 2 log 6) Simplify the expression log log 9 Leave your answer as an improper fraction. 7 2 Which of the following is smaller, 20 or 6 12? Solve for z: z = log loglog points each 6 Find the last digit in the sum of Evaluate: log 9 (8) log 10 (9) log 11 (10) log 096 (095) 1 8 Express log(1228) in terms of a and b given that log (07) = a and log (625) = b. Any term that does not contain a or b must be expressed as an integer or fraction. a b Solve for x: log(x) 1 = log(x 9) Solve for x and express your answer in root form. 5 5 x xxx... = 5
10 6 points each 11 Solve for x and express your answer in root form. (6 ) 8 = 108 ( 6) ( x) 12 A population of ants follows an increasing, exponential growth model, multiplying by 90 a factor of 15 5 every days. If the initial population was 10, what is the population 8 after days? Note that there are no such things as fractions of an ant when talking about population growth. 1 Solve for x in terms of a. 2 log b x = 2 log b (1 a) + 2 log b (1 + a) log b ( 1 a a) 2 1 Evaluate: log 2 81 log log e e 10 x = a Evaluate: log 81 ( 9 27 ) 1 8
11 Name: Units do not have to be included Log1 Contest Round 2 Mu Logs and Exponents Answer Key points each 1 Write in logarithmic form: 2 = 1 8 log 2 ( 1 8 ) = 2 Evaluate: log 5 0 log (log 2 log 6) Simplify the expression log log 9 Leave your answer as an improper fraction. 7 2 Which of the following is smaller, 20 or 6 12? Find the y-value of the point on the curve y = ln(x)dx when x =, given that when x = 1, y = 8 ln() points each 6 Find the last digit in the sum of Evaluate: log 9 (8) log 10 (9) log 11 (10) log 096 (095) 1 8 Express log(1228) in terms of a and b given that log (07) = a and log (625) = b. Any term that does not contain a or b must be expressed as an integer or fraction. a b Solve for x: log(x) 1 = log(x 9) Evaluate: log ( x ln()dx 0 + 1)
12 6 points each 11 Solve for x and express your answer in root form. (6 ) 8 = 108 ( 6) ( x) 12 A population of ants follows an increasing, exponential growth model, multiplying by 90 a factor of 15 5 every days. If the initial population was 10, what is the population 8 after days? Note that there are no such things as fractions of an ant when talking about population growth. 1 Solve for x in terms of a. 2 log b x = 2 log b (1 a) + 2 log b (1 + a) log b ( 1 a a) 2 1 Evaluate: log 2 81 log log e e 10 x = a Given that f(x) = ln e 2lne( x )lne x evaluate the derivative of f(x) at x = 2. That is, find f (2) 2
13 Log1 Contest Round 2 Logs and Exponents Solutions Mu Al Th Solution If a = b c, then log b a = c. Therefore, the logarithmic form of 2 = 1 8 is log 2 ( 1 8 ) = Since the quotient of two logs is the difference of the logs: log 5 0 log 5 8= log = log 55= 1 + log(7) 2log(7) = compared to Factor out 12 ( 2 ) compared to (2 ) 9 > 8 Therefore, 20 > 6 12 Since the log expression is equal to 0, it must be true that log 2 (log 12 x) = 1. If this is to be true as well, then log 12 x = 2. In exponential form, this is x = 12 2 = 1 5 ln(x)dx = x(ln(x) 1) + C 8 = 1(ln(1) 1) + C, ln(1) = 0 y = x(ln(x) 1) + 9 y = ln() + 6 when x = 5 5 z = log z 27 z =
14 6 6 6 For the base-2 number, the last digit follows the sequence 2,,8,6. Thus every power of for a base of 2 ends in 6. The next one is a is a multiple of so ends in a 6. For the base-7 number, the last digit follows the sequence 7,9,,1. This parallels the base- 2 number sequence so ends in a 7. Since = 1, the last digit in the sum is log 9 8 = log 8 log 9 log 10 9 = log 9 log 10 log = log 095 log 096 With the above change of bases, carrying out the multiplication sequence stated in the problem reduces the expression to log 8 log 096 = log 8 log 8 = log 8 log 8 = log(1228) = log(07 ) log(1228) = log(07) + log() log(1228) = a + log ( ) log(1228) = a + log(2500) log(625) log(1228) = a + log(25) + log(100) b log(1228) = a + log(625) b log(1228) = a log(625) b + 2 log(1228) = a b b + 2 log(1228) = a b log x + log(x 9) = 1 log(x(x 9)) = 1 x(x 9) = 10 1 = 10 x 2 9x 10 = 0 (x 10)(x + 1) = 0 x = 10 (positive roots only)
15 9 15 = 8ln(x) = 8 ln(x) 11 = ln(x) e 11 = x x = 1 e x lndx 0 log 81 = = 1 + ( 0 ) = x xxx... = 5 x = , ( ) = (2 6) (6 1 2) x2 ( ) = (6 2 ) (6 1 2) 6 16 = 6 8 (6 1 2) x2 x2 6 8 = x2 8 = 1 2 x 2 16 = x = x x = Population is equal to 10( t ), where t is equal to days. To find the population after days, divide by. Pop = 10 ( ) = 10 ( ) Pop = 10(2.5 ) = 10 ( 5 2 ) = 10 ( ) = Since fractions of an ant don t exist, the population is 90
16 1 1 1 Simplifying: 2 log b x = 2 log b ((1 a 2 ) log b ( 1 a a) 2 2 log b x = 2 log b ((1 a 2 ) 2log b ( 1 a2 ) a 2 log b x = 2 log b ( 1 a2 1 a 2 ) = 2 log b a a x = a By inspection, it is implied that a is restricted to 1 < a < +1 AND a 0. However, accept x = a as a correct answer. 1 1 Let a = log 2 81 a = log = log ( ) 1 2 a = log 6 a = 6 Let b = log log e e 10 b = log 2 (2 6 ) 1 10 = 2 10 b = Therefore, a b = ( 2x+6) (7 ) x = (7 2 ) x 7 2 2x+6 12x = 7 2x 8 8 1x = 2x 8 16x = 16 x = 1
17 15 f(x) = ln e 2lne( x )lne x = 2lne ( x )lne x f(x) = 2 (( x ) lne x ) = 6 ( 1 x ) x = 6x 1 2 f (x) = x 2 f (2) = (2) 2 = (8) 1 2 = 2 2 f (2) = log 81 ( ) = 1 2 log = 1 8
ln(9 4x 5 = ln(75) (4x 5) ln(9) = ln(75) 4x 5 = ln(75) ln(9) ln(75) ln(9) = 1. You don t have to simplify the exact e x + 4e x
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