Logarithms and Logarithmic Functions Coach Baughman November 20
Logarithms and Logarithmic Functions Coach Baughman November 20, 2003 Algebra II STAI 3
Objectives n n n The students will identify a logarithmic function. (Knowledge) (Mathematics, Algebra II, 6. a) The students will solve logarithmic expressions. (Application) (Mathematics, Algebra II, 6. b) The students will solve logarithmic functions. (Application) (Mathematics, Algebra II, 6. c) STAI 1, 11
John Napier n n Born in Edinburgh, Scotland, in 1550 Began education at St. Andrews University at the age of 13 Likely acquired mathematical knowledge at the University of Paris Died April 4, 1617 in Edinburgh, Scotland STAI 7
Logarithms n Definition: If b and y are positive where b 1, then the logarithm of y with base b (logby) is defined as logby = x if and only if bx = y. STAI 10
Special Logarithms n n logb 1 = 0 n Why? b 0 = 1 logbb = 1 n Why? b 1 = b The logarithm with base 10 is called the common logarithm. (log 10 or log) The logarithm with base e is called the natrual logarithm. (loge or ln) STAI 6, 23, 26
Examples n n Evaluate the expression log 381 n 3 x = 81 n 3 x = 3 4 nx = 4 Evaluate the expression log 1/28 n(1/2)x = 23 n(1/2)x = (1/2)-3 nx = -3 STAI 4, 19, 25
Logarithmic Functions n Exponential functions and logarithmic functions are inverses n n “undo” each other If g(x) = logbx and f(x) = bx, then g(f(x)) = logbbx = x and f(g(x)) = blogbx = x. STAI 23
Examples n Simplify the expression 10 log 2 =2 Simplify the expression x 9 x = log (32)x log 3 n 9 log 3 3 n n n = log 332 x = 2 x STAI 4, 19, 25
More Examples n Find the inverse of y = log 3 x n Use the definition of a logarithm n y = 3 x n Find the inverse of y = ln(x + 1) n n y = ln(x + 1) x = ln(y + 1) (switch x and y) ex = y + 1 (write in exponential form) ex – 1 = y (solve for y) STAI 4, 19, 25
Assessment 1. 2. 3. 4. 5. 6. Write log 7 b = 13 in exponential form. Write 43 = 64 in logarithmic form. Solve the equation logx(1/32) = -5. Simplify log 5252. Evaluate log 4256. Find the inverse of y = ln(2 x – 5) STAI 25, 33, 36
Closing Questions What did we learn about today? n Can anyone tell me the definition of a logarithm? n Where might you use logarithms? n STAI 10, 26
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