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Introduction to Logarithmic Functions

Learn about logarithmic functions and their properties, including writing equations in logarithmic form, evaluating logarithmic expressions, solving logarithmic equations, and using logarithmic properties. Complete the assigned homework problems.

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Introduction to Logarithmic Functions

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  1. Warm-Up 4/30 Answer: $62,426.36 $60,900.52

  2. 11.4 Logarithmic Functions x = by logarithm y = logbx The inverse of y = bx is _______ The function x = by is called a___________ It is usually written ______________ Read “ y equals log base b of x” Logarithmic functions are the inverse of exponential functions

  3. Definition:y= logbx if and only if x=by EX1:Write in exponential form a) log273 = 1/3 b) log164 = ½ • “b” can’t be 1 and it must be positive Answers: EX2: Write each equation in logarithmic form a) 210 = 1024 b) 2-3 = 1/8 Answers: log2 1024 = 10 b) log2 1/8 = -3

  4. Ex3 Evaluate: log51/625 • 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 log51/625 = -4

  5. Ex 4: evaluate log432 • Think 4 to the what equals 32 • Nothing – dang it • Re-write: 4x = 32 • Get the bases the same: (22)x = 25 • Bases are same so just set exponents equal to each other • 2x = 5 • X = 2.5

  6. 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…

  7. Ex 6 Solve:log10 (2x+5) = log10(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.

  8. Ex 7: Solve log3(4x+5) – log3(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 + 5 • -22x=-22 • x=1

  9. Ex8: log3(x+2)+log3(x-6) = 2 log3(x+2)(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 # 32 = (x+2)(x – 6) 9 = x2 – 4x – 12 0 = x2 – 4x – 21 (x – 7)(x + 3)=0 x = 7 x = -3

  10. Ex 9: ½ log8(x+1) – ½ log825 = log84 • Use your properties to write as a single log on each side Subtraction means division Cross multiply and solve Square both sides

  11. Summary: • Homework: pg 723 # 20-52

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