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Name: Date: 5.1 notes on Logarithmic Functions The inverse of the exponential function is the logarithmic function. We write this as: The inverse of the natural exponential function is the natural logarithmic function. We write this as: Relationships between logb x and b x Since exponential functions and logarithmic functions are inverses of each other, for b > 0 and b 1, we have the following relationships: a. b. c. d. Page 1 Graph of: f ( x) logb x , with b>1 f ( x) logb x , with 0<b<1 Algebraic Rules for Logarithmic Functions: For all values of x, y, b, and a for which these expressions are defined, we have a. e. b. f. c. g. d. h. Page 2 Examples: Determine which of the given functions are exponential. Write each exponential function in the form f ( x) Ab x 1. f ( x) 4(1.2) x 5 2. f ( x) 3.1x 7.4 3. f ( x) 2 x31 x Use the fact that y e x and y ln x are inverses to simplify the following expressions. 1. eln 3 2. e 3ln x 3. ln(e3 x ) Write the function f ( x) 2 x in the form f ( x) ekx , and write the function g ( x) e2 x in the form g ( x) b x . Page 3 Calculate each of the following by hand, without a calculator. 1. log 2 64 2. log6 3 log 6 12 3. 2log3 6 log3 4 Solve each of the following equations. 1. 3.25(1.72) x 1000 2. 3x 9 x2 Challenge Problem: 9(9x ) 10(3x ) 1 0 Homework: Properties of Logarithms Worksheet #1-23odd Page 4