4. Find x, log 4 32 = x. 5. ln e ln ln e. 8. log log log 3 243

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1 Session #5 Logarithms What is a log? it is the inverse of an exponential equation. When you take the inverse of an exponential equation with a base of e, you have a natural logarithm, written as ln. 1. Evaluating Logs The word log really means What is the exponent? SO a question like log 2 64 translates into What is the exponent on a base of 2 that results in an answer of 64? When no base is written, it is understood to be a base of 10. Ex. Evaluate 1. log log Find x, log x 25 = 2 4. Find x, log 4 32 = x 5. ln e 7 6. ln 1 7. ln e 8. log log log

2 1. Log Rules 1. When you are multiplying with the same base you add the exponents, so if you are multiplying values with the sam base, you add the logs. log a xy = log a x + log a y 2. When you are dividing with the same base you subtract the exponents, so if you are dividing values with the same base, you subtract the logs. log a( x/y) = log a x log a y When you are raising a power to a power, you multiply the exponents together, so when you are have the log of something raised to a power you multiply the power and the log (in other words, you bring it down front). log a x y = ylog a x Examples 1. If log 2 = x and log 3 = y, write the following in terms of x and y. a. log 9 b. log 18 c. log Write as a single log: log x = log 3 (2loga + ½ logb) 3. If x = 4x 3, then the logx = y 5 4. log 2 64 log log 4 x log 4 (x + 3) = 1 5. log 2 x + log 2 (x 4) = 5 2

3 Exponent Rules and Log Rules 3

4 7. Solve for x: log 4 (x 2 + 3x) log 4 (x + 5)= 1 8. The expression log 10 x+2 log10 x is equivalent to 9. If log a = 2 and log b = 3, what is the numerical value of log ( a)/b 3? 10. Express as a single natural logarithm (2lnx + lny) - ½ lnz 11. Expand using log rules: ln[(x - 4)(2x + 5)] 2 4

5 3. Solving Log Equations What do you do? 1. When the exponent is a variable and the bases are not powers of the same base, you will take the logarithm of both sides of the equation. 2. Isolate what is being raised to the power before taking the log/ln of both sides. 1. Depreciation (the decline in cash value) on a car can be determined by the formula V = C(1 r) t, where V is the value of the car after t years, C is the original cost, and r is the rate of depreciation. If a car s cost, when new, is $21,000, the rate of depreciation is 24%, and the value of the car is now $5,500, how old is the car to the nearest tenth of a year? 2. The amount A, in milligrams, of a 25 milligram dose of a drug remaining in the body after t hours is given by the formula A = 25(.73) t. Find, to the nearest tenth of an hour, how long it takes for half of the drug dose to be left in the body. 3. Growth of a certain strain of bacteria is modeled by the equation G = A(2.7) 0.584t, where: G = final number of bacteria A = initial number of bacteria t = time (in hours) In approximately how many hours will 9 bacteria first increase to 623 bacteria? Round your answer to the nearest hour. 8 hours at 7 hours it will not have reached 623 yet. 5

6 4. An archaeologist can determine the approximate age of certain ancient specimens by measuring the amount of carbon 14, a radioactive substance, contained in the specimen. The formula used to determine the age of a specimen is A = A 0 2 t/5760 where A is the amount of carbon 14 that a specimen contains, is the original amount of carbon 14, t is time, in years, and 5760 is the half life of carbon 14. A specimen that originally contained 201 milligrams of carbon 14 now contains 83 milligrams of this substance. What is the age of the specimen, to the nearest hundred years? 5. Solve for x to the nearest thousandth: e 0.24x =

7 5. A hotel finds that its total annual revenue and the number of rooms occupied daily by guests can best be modeled by the function R = 3log(n n), n > 0 where R is the total annual revenue, in millions of dollars, and n is the number of rooms occupied daily by guests. The hotel needs an annual revenue of $12 million to be profitable. Graph the function on the accompanying grid over the interval 0 < n < 100 Calculate the minimum number of rooms that must be occupied daily to be profitable. 7

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9 6. Solve for x: ln(3x + 2) + ln(x - 1) = ln(3x 2 + x - 6) 7. Solve for x in terms of e: 3ln(2x) = Show, using the log rules, that ln 2 = ln (½ ) On January 1, 2010, an initial investment of $10,000 is made. The money will remain untouched in the account. The interest is compounded continuously at a rate of 4.1% for the li of the investment. During what year will the investment reach $25,000? 9

10 6. Mouthwash manufacturers are constantly testing chemicals on bacteria that thrive on human saliva. The death of the bacteria exposed to Antigen 223 can be represented by the function P(t) = 2,000e -0.37t, where P(t) represents the number of bacteria from a population of 2000 survivors after t minutes. a. Determine the number of bacteria surviving after 8 minutes. b. Determine the number of minutes, to the nearest tenth of a minute, necessary to kill 90% of the bacteria. 10

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