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Name ___________________________________________ Date _____________________ Class ____________________ 4.3: Introduction to Logarithmic Functions Unit 4: Exponential & Logarithmic Functions Big Idea #1: Exponential equations can also be written as logarithmic equations. Big Idea #2: The logarithm is equal to the exponent in the exponential equation. be = n logb n = e 1. Complete the table. Exponential Equation 33 = 27 Logarithmic Equation log3 27 = 3 5-1 = 0.2 80 = 1 log4 16 = 2 2. Explain why the logarithm of 1 with base b is always 0. _________________________________________________________________________________________________ Write each exponential equation in logarithmic form. 3. 37 = 2187 ___________________________ 4. 122 = 144 ___________________________ 5. 53 = 125 ___________________________ Write each logarithmic equation in exponential form. 6. log10 100,000 = 5 ___________________________ 7. log4 1024 = 5 ___________________________ 8. log9 729 = 3 ___________________________ Evaluate without a calculator. 9. log 1,000,000 ___________________________ 12. log4 16 ___________________________ 10. log 10 ___________________________ 13. log8 1 ___________________________ 11. log 1 ___________________________ 14. log5 625 ___________________________ Solve for x in each without a calculator. 15. logx 16 = 2 ___________________________ 18. log x = –2 ___________________________ 16. logx 27 = 3 ___________________________ 19. log5 13 = log5 (19 – 2x) ___________________________ 17. log7 x = 3 ___________________________ 20. log2 8x ___________________________ Name ___________________________________________ Date _____________________ Class ____________________ Big Idea #3: A logarithmic function is the inverse of an exponential function. You can identify an inverse function by comparing its graph to the graph of the original function. The two graphs are a reflection of each other across the line y = x. Exponential function: f (x) = bx The base b is any number greater than 1. Example: g(x) = 3x The domain is all real numbers. The range is all positive numbers. Logarithm function: f-1(x) = logb x Use the same base to find the inverse function. The inverse of g(x) is g-1 (x) = log3 x. In the inverse function, the domain and range are switched. 21. Find the inverse function of f (x) = 4x. 22. a. Find the inverse function of g(x) = 1x 5 _______________________ b. What are the domain and range of g(x) and g-1(x)? Domain of g(x): __________________ Range of g(x): Domain of g -1(x): __________________ Range of g-1(x): __________________ __________________ Without a calculator, use the given x-values to graph each function. Then graph its inverse. Describe the domain and range of both functions. After, check the graphs using a calculator. x 23. f (x) = 0.1x; x = -1, 0, 1, 2 5 24. f ( x ) = ; x = -3, -2, -1, 0, 1, 2, 3 2 Original: ____________________________________ Original: ____________________________________ Inverse: _____________________________________ Inverse: _____________________________________ Name ___________________________________________ Date _____________________ Class ____________________ 4.3 INTRODUCTION TO LOGARITHMS 2 1.4 = 16; log5 0.2 = −1; log6 1 = 0 24. Domain: {x|x > 0};range: all real numbers 2. logb 1 = 0 is the same as b0 = 1 and any number to the 0 power is 1. 3. log3 2187 = 7 4. log12 144 = 2 5. log5 125 = 3 6. 105 = 100,000 7. 45 = 1024 8. 93 = 729 9. 6 10. 1 11. 0 12. 2 13. 0 14. 4 15. x = 4 16. x = 3 17. x = 343 18. x = 0.01 19. x = 3 20. 3x 21. f −1(x) = log4 x 22. a. g −1(x) = log1/5 x b. Domain of g(x) is all real numbers Challenge 25. The hydrogen ion concentration in moles per liter for a certainbrand of tomato-vegetable juice is 0.000316. a. Write a logarithmic equation for the pH of the juice. b. What is the pH of the juice? range of g(x) is y > 0 domain of g−1(x) is x > 0 −1 range of g (x) is all real numbers. 25. a. pH = -log(0.000316) b. 3.5 23. Domain: {x|x > 0}; range: all real numbers Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor. Holt McDougal Algebra 2