Warm-Up 4/30 Answer: $62, $60,900.52

11.4 Logarithmic Functions  The inverse of y = b x is _______  The function x = b y is called a___________  It is usually written ______________  Read “ y equals log base b of x”  Logarithmic functions are the inverse of exponential functions x = b y logarithm y = log b x

Definition: y= log b x if and only if x=b y “b” can’t be 1 and it must be positive EX1: Write in exponential form a) log 27 3 = 1/3b) log 16 4 = ½ Answers: EX2: Write each equation in logarithmic form a) 2 10 = 1024b) 2 -3 = 1/8 Answers: log = 10b) log 2 1/8 = -3

Ex3 Evaluate: log 5 1/625 This is a number, its an operation The answer to a log will be an exponent Think 5 to the what power is 1/625 Since it is a fraction the exponent will be negative 5 4 = 625 so 5 –4 =1/625 So log 5 1/625 = -4

Ex 4: evaluate log 4 32 Think 4 to the what equals 32 Nothing – dang it Re-write: 4 x = 32 Get the bases the same: (2 2 ) x = 2 5 Bases are same so just set exponents equal to each other 2x = 5 X = 2.5

Since a log is inverse of an exponent it follows the exponent rules… m and n are positive numbers, b is a positive number other than 1 and p is any real number… Property Definition Product log b mn =log b m +log b n Quotient log b m/n = log b m – log b n Power log b m p = p(log b m) Power of equality If log b m=log b n then m=n

Ex 6 Solve: log 10 (2x+5) = log 10 (5x-4) Which property can I use? Power of equality… the bases are the same and they are equal so 2x+5 = 5x – 4 easy 9 = 3x x = 3 are they all this easy – of course not you silly geese.

Ex 7: Solve log 3 (4x+5) – log 3 (3 – 2x) = 2 Don’t have logs on both sides so we can’t use the equality property. Always try to simplify – subtraction, write it as a quotient Re-write using definition of logs now solve /cross multiply 27 – 18x = 4x x=-22 x=1

Ex8: log 3 (x+2)+log 3 (x-6) = 2 Write as a single log: Use log properties: No logs on both sides Write in exponential form Solve: This is a Quadratic You should know how to solve CHECK in original equation You might need to eliminate an answer Can’t take the log of a neg # log 3 (x+2)(x – 6)=2 3 2 = (x+2)(x – 6) 9 = x 2 – 4x – 12 0 = x 2 – 4x – 21 (x – 7)(x + 3)=0 x = 7 x = -3

Ex 9: ½ log 8 (x+1) – ½ log 8 25 = log 8 4 Use your properties to write as a single log on each side Subtraction means division Cross multiply and solve Square both sides

Summary: Homework: pg 723 # 20-52