Log1 Contest Round 2 Theta Logarithms & Exponents. 4 points each

Transcription

1 5 Log Contest Round Theta Logarithms & Eponents Name: points each Simplify: log log65 log6 log6log9 log5 Evaluate: log Find the sum:... A square has a diagonal whose length is feet, enclosed by the square. log log 5 Evaluate: log 5 log 6 log 9log feet. Find the area, in square 5 points each 6 Given that f, find the value of f. ln e Simplify: log 9 e ln5 If a and, find the numerical value of 5a a. 6 9 The number of bacteria in a culture grows eponentially according to the equation N ln.5t, where N is the initial population of the bacteria culture and N is N the population of the culture after t hours. To the nearest hour, how long does it take for the culture to double its initial population? A bank offers an interest-bearing savings account at % annual interest, compounded semiannually. If I opened this type of account at this bank with an initial deposit of $, and if I make no deposits or withdrawals from the account, how much would I have in my account after two years (rounded down to the nearest cent)?

2 Solve for : log9 log6 log 6 points each Find the sum of all comple roots of the equation : 9 5 Is the real solution to the equation greater than (>), less than (<), or equal to (=)? Write as your answer >, <, or =. If the real solution to the equation log y, find the value of y can be written in the form 5 Find the sum of the common logarithms of all positive integral factors of,.

3 5 Log Contest Round Alpha Logarithms & Eponents Name: Simplify: log log65 log6 log6log9 log5 Evaluate: log points each 5 Solve for the greater value of : 56 6 A square has a diagonal whose length is feet, enclosed by the square. Evaluate: 5 i e log log feet. Find the area, in square 5 points each 6 Given that f, find the value of Given that log a, log5 b, and log c terms of a, b, and c. If a and 6 f., find, find the numerical value of 5a a. log 5 log log 5 log 9 in The number of bacteria in a culture grows eponentially with growth constant.5/hour, where the eponential function has base e. To the nearest hour, how long does it take for the culture to double its initial population? A bank offers an interest-bearing savings account at % annual interest, compounded semiannually. If I opened this type of account at this bank with an initial deposit of $, and if I make no deposits or withdrawals from the account, how much would I have in my account after two years (rounded down to the nearest cent)?

4 Solve for : log9 log6 log 6 points each Find the sum of all real roots of the equation : 5 9 Find all real solutions to the equation: If the real solution to the equation log y, find the value of y can be written in the form 5 Find the sum of the common logarithms of all positive integral factors of,.

5 5 Log Contest Round Mu Logarithms & Eponents Name: Simplify: log log65 log6 log6log9 log5 points each Solve for the greater value of : 56 6 Find the slope of the tangent to the graph of y log e at the point whose y- coordinate is log e. Evaluate: 5 i e 5 Given that f, find the value of f. 6 Find the slope of the tangent to the graph of Given that log a, log5 b, and log c terms of a, b, and c. If a and 6 5 points each at the point, e. sin y e log, find, find the numerical value of 5a a. log 5 log log 5 log 9 in The number of bacteria in a culture grows eponentially with growth constant.5/hour, where the eponential function has base e. To the nearest hour, how long does it take for the culture to double its initial population? A bank offers an interest-bearing savings account at % annual interest, compounded either annually or semiannually. I opened two accounts, one with each type of compounding. I made an initial deposit of $ into each account, and I made no deposits to or withdrawals from either account for two years. If the amount of money in each account is rounded down to the nearest cent, then how much more money would I have in the semiannually compounded account after those two years?

6 Solve for : log9 log6 log 6 points each Find the sum of all imaginary roots of the equation : 5 9 Find the sum of the common logarithms of all positive integral factors of,. Evaluate: cos d 5 If the area enclosed by the graphs of y, y log,, and can be written in the form log y, find the value of y.

7 5 Log Contest Round Theta Logarithms & Eponents Name: points each Simplify: log log65 log6 log6log9 log5 5 or.5 Evaluate: log 99 Find the sum:... A square has a diagonal whose length is feet, enclosed by the square. log log feet. Find the area, in square 5 Evaluate: log 5 log 6 log 9log 5 points each 6 Given that f, find the value of f. ln e Simplify: log 9 e ln5 If a and, find the numerical value of 5a a. 6 9 The number of bacteria in a culture grows eponentially according to the equation N ln.5t, where N is the initial population of the bacteria culture and N is N the population of the culture after t hours. To the nearest hour, how long does it take for the culture to double its initial population? A bank offers an interest-bearing savings account at % annual interest, compounded semiannually. If I opened this type of account at this bank with an initial deposit of $, and if I make no deposits or withdrawals from the account, how much would I have in my account after two years (rounded down to the nearest cent)? $.

8 6 points each Solve for : log9 log6 log 5 Find the sum of all comple roots of the equation : 9 Is the real solution to the equation greater than (>), less than (<), or equal to (=)? Write as your answer >, <, or =. If the real solution to the equation log y, find the value of y can be written in the form > or.5 5 Find the sum of the common logarithms of all positive integral factors of,. 9

9 5 Log Contest Round Alpha Logarithms & Eponents Name: Simplify: log log65 log6 log6log9 log5 points each 5 or.5 Evaluate: log 99 5 Solve for the greater value of : 56 6 A square has a diagonal whose length is feet, enclosed by the square. Evaluate: log log feet. Find the area, in square 5 i e 5 points each 6 Given that f, find the value of Given that log a, log5 b, and log c terms of a, b, and c. If a and 6 f., find log 5 log log 5 log 9 in 9 5 a b a c, find the numerical value of 5a a. 9 The number of bacteria in a culture grows eponentially with growth constant.5/hour, where the eponential function has base e. To the nearest hour, how long does it take for the culture to double its initial population? A bank offers an interest-bearing savings account at % annual interest, compounded semiannually. If I opened this type of account at this bank with an initial deposit of $, and if I make no deposits or withdrawals from the account, how much would I have in my account after two years (rounded down to the nearest cent)? $.

10 6 points each Solve for : log9 log6 log Find the sum of all real roots of the equation : 5 9 Find all real solutions to the equation: If the real solution to the equation log y, find the value of y can be written in the form log or equiv. 5 Find the sum of the common logarithms of all positive integral factors of,. 9 or.5

11 5 Log Contest Round Mu Logarithms & Eponents Name: Simplify: log log65 log6 log6log9 log5 points each 5 or.5 Solve for the greater value of : 56 6 Find the slope of the tangent to the graph of y log e coordinate is log e. at the point whose y- or.5 Evaluate: 5 i e 5 Given that f, find the value of f. 6 Find the slope of the tangent to the graph of Given that log a, log5 b, and log c terms of a, b, and c. If a and 6 5 points each at the point, e. ln or equiv. sin y e log, find log 5 log log 5 log 9 in 9 5 a b a c, find the numerical value of 5a a. 9 The number of bacteria in a culture grows eponentially with growth constant.5/hour, where the eponential function has base e. To the nearest hour, how long does it take for the culture to double its initial population? A bank offers an interest-bearing savings account at % annual interest, compounded either annually or semiannually. I opened two accounts, one with each type of compounding. I made an initial deposit of $ into each account, and I made no deposits to or withdrawals from either account for two years. If the amount of money in each account is rounded down to the nearest cent, then how much more money would I have in the semiannually compounded account after those two years? $. or cents

12 6 points each Solve for : log9 log6 log Find the sum of all imaginary roots of the equation : 5 9 Find the sum of the common logarithms of all positive integral factors of,. 9 Evaluate: cos d 5 If the area enclosed by the graphs of y, y log,, and can be written in the form log y, find the value of y. 6 e

13 5 Log Contest Round Logarithms & Eponents Solutions Mu Al Th Solution log log65 log6 5log log5 log6 5 5 log6 log9 log5 log6 6log log log log log log 99 Each term in the series equals, so we only need to determine how many terms are in the series. Since the first has no eponents and the last has eponents, there must be a total of eight s in the series, so the sum is , so the greater solution is. Since log y e, y e log ' log ln =. Further, if the y-coordinate is log e, then, making the slope of the tangent. log log, so because this is the diagonal length log log log of the square, the sides of the square must be, making the enclosed area. 5 Since log, the first three numbers of the product, which are all positive real numbers, have a product being multiplied by, so the product is. 5 5 i e cos 5 isin 5 i f f 5 f f 6 sin sin Since y e log, y' e cos, making the slope of the tangent at ln the given point y' e. ln ln ln ln5 e ln5 e e 5 log 9 log 9 log 9 log log 5log log 5log log log5 log loglog5 ab log log log5 log log. ac log5 log log5 log9 log log5 log log log 9 log5 log log e

14 9 9 Since a a 6 5a, a. Since, 6 5a. From the first 5a equation, a 6 a. Plugging this into the second equation, a 5a a. Plugging this value into the first equation,. Therefore, 9. 9 a 9 9 Plugging in N N to the equation yields ln.5t t ln. Using the approimation ln.69, the amount of time required to double the initial population is t ln.69.6 hours, which rounded to the nearest hour means it takes hours to double. N The equation for this growth model would be ln.5t, where N is the initial N population of the bacteria culture and N is the population of the culture after t hours (units suggested by the units given in the problem). Plugging in N N to the equation yields ln.5t t ln. Using the approimation ln.69, the amount of time required to double the initial population is t ln.69.6 hours, which rounded to the nearest hour means it takes hours to double...6, so rounded down to the nearest cent, the account has $.. The amount in the account after two years would be A. The amount in the semiannually compounded account after two years would be. A..6, so rounded down to the nearest cent, that account has $.. The amount in the annually compounded account after two. years would be A..6, so that account has $.6. Therefore, the semiannually compounded account has $. $.6 $. more in it. log log log log9 log6 log , so all three roots of the polynomial 9 are distinct, and the sum of the roots is , and the roots of the quadratic are imaginary, so the sum of the real roots is , and the roots of the quadratic are imaginary, so the sum of the imaginary roots is.

15 Since 5 5 Since y is a continuous increasing function, y 6, and solution must be greater than (since we are looking to solve y ). y, the. However, there is no real solution to, so the only real solution is log , and there is no real solution to the second factor equals. Therefore, 6 6 log, so 5 5 5, 5 y., it has positive integral factors. Further, each positive integral factor can be paired with the factor whose product with it is, when combining the logarithms for eample, log log, log, ; loglog5, log, ; and so on. Since there are 6 positive integral factors, and since, is not a perfect square, there are such pairs of distinct positive integral factors, each of whose corresponding sum is log, 5. Therefore, the sum is 5 9. Using Wallis integral formula, OR cos d 6 5 cos d cos d cos cos d cos cos cos d cos d sin sin C sin sin c, so cos d sin sin sin sin. 6 ln log d ln Based on the graphs, the area would be d ln ln ln d ln d ln ln ln ln ln e e, so y. ln ln ln ln ln ln ln lne ln ln e log