Math 147 Section 5.2. Application Example

Transcription

1 Math 147 Section 5.2 Logarithmic Functions Properties of Change of Base Formulas Math 147, Section Application Example Use a change-of-base formula to evaluate each logarithm. (a) log 3 12 (b) log 9 (0.35) Math 147, Section

2 or logs are inverse concept from exponentials. We know 2 5 = 32. Another way of saying this is. Math 147, Section Verbal Algebraic Example y = log a x means y is the to which the base a must raised to get x. y = log a x means. log 2 32 = 5 means. Math 147, Section

3 Write 125 = 5 3 in log form: The base is 5, the exponent (or log) is 3, so Write log 3 (1/9) = 2 in exponential form: The base is 3, and the log (or exponent) is 2 so this is same as Math 147, Section Find x: 3 = log 2 x. Base is 2, log (or exponent) is 3 so x =. Math 147, Section

4 Find y: y = log 3 (1/81) then, since 3 4 = 81,, so y = Math 147, Section Graph y = log 3 x by plotting points of x = 3 y x = 3 y y /3 1 1/9 2 1/27 3 Math 147, Section

5 Graph y = log 3 x using calculator. Y= MATH A 3!" X,T,θ,n ENTER Math 147, Section Graphs of Logarithmic functions: y= log a x (a>1) x-intercept = Domain: Range: Asymptote: Math 147, Section

6 We have three types of logs we will see: Logs with base written simply as log with no base specified. Logs with base written simply as ln. Logs with base other than 10 or e, say written as log a Math 147, Section Doubling Time for an Investment For P (principal) invested for t years at rate r, compounded continuously, the future value (S) is We want the investment to double or S = 2P so we solve 2P = Pe rt dividing by P gives: 2 = e rt x = a y means log a x = y Changing 2 = e rt to ln form we get solve this for t we get t = Math 147, Section

7 Doubling Time for an Investment t = (ln 2)/r = /r For a 10% rate this becomes 0.693/.10 = 69.3/10 = 6.93 This is the basis for the - to determine how long it take an investment at r%, compounded continuously, to double, divide 70 by the rate: 70/r. A 5% an investment should double in approximately 70/5 or 14 years. Math 147, Section y = log x vs. y = 10 x By the definition of logarithms, we see that y = log a x and a y = x or y = a x are related. x y = log x x y = 10 x / /10 Math 147, Section y = log x and y = 10 x are 7

8 y = log x vs. y = 10 x f(x) and g(x) are if for each that satisfies one, satisfies the other. In this case, since x- and y-coordinates are interchanged, the graphs of these are reflected across the line y = x. Math 147, Section Properties of : Property I log a a x = x a > 0, a 1 y = log a a x means a y = a x so y = x or Math 147, Section

9 Properties of : Property I log a a x = x a > 0, a 1 Examples: log = _ ln e 7 = _ Math 147, Section Properties of : Property I Special Cases of Property I: log a a x = x Since a 1 = a, log a a = 1 Since a 0 = 1, log a 1 = 0 Math 147, Section

10 Properties of : Property II a > 0, a 1 Let Take log base a of each side: By Property I: Or log a y = log a x, so y = x or Math 147, Section Properties of : Property II Examples: e ln x = _ Math 147, Section

11 Properties of : Property III If a > 0, a 1, and M and N are positive real numbers then: log a (MN) = log a M + log a N Let u = log a M and v = log a N, then a u = M and a v = N So log a (MN) = log a (a u a v ) = log a (a u+v ) By Property I: log a (a u+v ) = u + v = log a M + log a N So: log a (MN) = log a M + log a N Math 147, Section Properties of : Property III log a (MN) = log a M + log a N Example: log 2 4 = _ and log 2 16 = _ so log 2 (64) = log 2 (4x16) = Math 147, Section

12 Properties of : Property IV a > 0, a 1, and M and N are positive real numbers: log a (M/N) = log a M log a N Let u = log a M and v = log a N then a u = M and a v = N So: log a (M/N) = log a (a u /a v ) = log a (a u v ) By Property I: log a (a u v ) = u v = log a M log a N So: log a (M/N) = log a M log a N Math 147, Section Properties of : Property IV log a (M/N) = log a M log a N Example: log 3 (9/27) = = _ _ = Math 147, Section

13 Properties of : Property V a > 0, a 1, and M is a positive real number and N is any real number: log a (M N ) = N log a M Let u = log a (M N ) then a u =M N (a u ) 1/N = (M N ) 1/N a u/n = M log a M = log a (a u/n ) = u/n N log a M = u so N log a M = log a (M N ) Math 147, Section Properties of : Property V log a (M N ) = N log a M Example: log 3 (9 2 ) = = = _ Math 147, Section

14 Change of Base Formulas Many calculators only can use log (base 10) and ln. So it would be nice to be able to express log b in terms of log or ln. Math 147, Section Change of Base Formulas y = log b x means b y = x Take the base a log of both sides: log a b y = log a x y log a b = log a x Math 147, Section

15 Change of Base Formulas If a > 0, b > 0, a 1, b 1, then In general: For base e: For base 10: Math 147, Section Change of Base Formulas Evaluate log 6 24: Math 147, Section

16 Solve for x: Math 147, Section Simplify: log 3 27 Math 147, Section

17 log a x = 3.1; log a y =1.8; log a z =2.7 log a (xy) = Math 147, Section Example The Richter scale is a measure of intensity of an earthquake. The magnitude of an earthquake of intensity I is R= log (I/I 0 ) where I 0 is a certain minimum intensity used for comparison. How much stronger is an earthquake of magnitude 7 on the Richter scale than one of magnitude 5? Math 147, Section

18 Example How much stronger is an earthquake of magnitude 5 on the Richter scale than one of magnitude 3? Math 147, Section Example Loudness (in decibels) = 10 log (I/I 0 ), where I 0 is the threshold of hearing for the average ear. The sound that causes pain is about times I 0. How many decibels is this? How many decibels is the average threshold of hearing? Math 147, Section

19 Example Doubling time t, in years for an investment of r% compounded n times per year is Find t for a rate r of 7.2% compounded monthly. Math 147, Section Example Find t for a rate r of 7.2% compounded monthly. Math 147, Section

20 Example Strongest earthquake, 9.5 on Richter scale, hit Chile in In 2010 Chile had quake of 8.8 on Richter. How many times stronger was the 1960 quake? Math 147, Section Application Example Use a change-of-base formula to evaluate each logarithm. (a) log 3 12 (b) log 9 (0.35) Math 147, Section

21 Common Mistakes The log of a sum is NOT the sum of the logs. The sum of the logs is the log of the product. The log of a sum cannot be simplified. log a (x + y) log a x + log a y Math 147, Section Common Mistakes The log of a difference is NOT the difference of the logs. The difference of the logs is the log of the quotient. The log of a difference cannot be simplified. log a (x y) log a x log a y Math 147, Section

22 Common Mistakes An exponent on the log is NOT the coefficient of the log. Only when the argument is raised to a power can the exponent be turned into the coefficient. When the entire logarithm is raised to a power, then it can not be simplified. (log a x) r r * log a x Math 147, Section Common Mistakes The log of a quotient is not the quotient of the logs. The quotient of the logs is from the change of base formula. The log of a quotient is the difference of the logs. log a (x/y) ( log a x )/( log a y ) Math 147, Section