Logarithms and Logarithmic Functions Coach Baughman November 20, 2003 Algebra II STAI 3

Objectives 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 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 Definition: If b and y are positive where b1, then the logarithm of y with base b (log b y) is defined as log b y = x if and only if b x = y. STAI 10

Special Logarithms log b 1 = 0 Why? b 0 = 1 log b b = 1 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. (log e or ln) STAI 6, 23, 26

Examples Evaluate the expression log x = 81 Evaluate the expression log 1/2 8 (1/2) x = 8 3 x = 3 4 x = 4 (1/2) x = 2 3 (1/2) x = (1/2) -3 x = -3 STAI 4, 19, 25

Logarithmic Functions Exponential functions and logarithmic functions are inverses “undo” each other If g(x) = log b x and f(x) = b x, then g(f(x)) = log b b x = x and f(g(x)) = b log b x = x. STAI 23

Examples Simplify the expression 10 log2 10 log2 = 2 Simplify the expression log 3 9 x log 3 9 x = log 3 (3 2 ) x = log 3 3 2x = 2x STAI 4, 19, 25

More Examples Find the inverse of y = log 3 x Use the definition of a logarithm y = 3 x Find the inverse of y = ln(x + 1) y = ln(x + 1) x = ln(y + 1) (switch x and y) e x = y + 1 (write in exponential form) e x – 1 = y (solve for y) STAI 4, 19, 25

Assessment 1. Write log 7 b = 13 in exponential form. 2. Write 4 3 = 64 in logarithmic form. 3. Solve the equation log x (1/32) = Simplify log Evaluate log Find the inverse of y = ln(2x – 5) STAI 25, 33, 36

Closing Questions What did we learn about today? Can anyone tell me the definition of a logarithm? Where might you use logarithms? STAI 10, 20, 26