Section 4.2 d i E Logarithmic Functions a l l u d Dr. Abdulla Eid b A College of Science . r D MATHS 103: Mathematics for Business I Dr. Abdulla Eid (University of Bahrain) Logarithms 1 / 8
1 - The Logarithmic Functions Recall: The exponential function is d i E f ( x ) = a x , a > 0, a � = 1 a l l u The general shape of y = a x is either d b A . r D Domain = ( − ∞ , ∞ ) . Range = ( 0, ∞ ) . Dr. Abdulla Eid (University of Bahrain) Logarithms 2 / 8
Question: Is f ( x ) has an inverse? Why? Answer: Yes, by the horizontal line test and the graph of the inverse function f − 1 ( x ) is either d i E a l l u d f − 1 ( x ) is called logarithmic function base a and it is denoted by b A f − 1 ( x ) = log a x . r D Note: (The fundamental equations) 1 f ( f − 1 )( x ) = x , so we have a log a x = x . 2 f − 1 ( f ( x )) = x , so we have log a a x = x . Dr. Abdulla Eid (University of Bahrain) Logarithms 3 / 8
2 - Exponential and Logarithmic forms We have the following log a x = y if and only if x = a y � �� � � �� � d exponential form logarithmic form i E a l Example l u d Convert from logarithmic form to exponential form and vice versa. b 1 3 2 = 9 ⇐ A ⇒ 2 = log 3 9. 2 log 2 1024 = 10 ⇐ ⇒ 1024 = 2 10 . . r D 3 e − 5 = y ⇐ ⇒ − 5 = log e y . 2 2 4 8 3 = 4 ⇐ ⇒ 3 = log 8 4. 5 log 2 1 32 = 2 − 5 . 1 32 = − 5 ⇐ ⇒ 6 3 0 = 1 ⇐ ⇒ 0 = log 3 1. Dr. Abdulla Eid (University of Bahrain) Logarithms 4 / 8
Exercise Convert from the exponential form into logarithmic form and vice versa 1 log 7 x = 5. √ 2 = 1 2 log 2 2 . 3 9 3 = 729. d √ i E 1 4 5 3 = 3 5. a l l u d b A . r D Dr. Abdulla Eid (University of Bahrain) Logarithms 5 / 8
Example Solve for x the equation log 3 x = 4. Solution: We convert it into exponential form to get x = 3 4 = 81 d i E Solution set = { 81 } . a l Example l u d Solve for x the equation log x 4 = 1 2 . b A Solution: We convert it into exponential form to get . r D 1 4 = x 2 4 2 = ( x 1 2 ) 2 16 = x Solution set = { 16 } . Dr. Abdulla Eid (University of Bahrain) Logarithms 6 / 8
Example Solve for x the equation log 4 x = − 4. Solution: We convert it into exponential form to get d i 1 x = 4 − 4 = E 256 a l l Solution set = { 1 u 256 } . d b Exercise A Solve for x the equations . r D 1 log 5 x = 3. 2 log 3 1 = 0. 3 log a 1 = 0. 4 log x ( 2 x + 8 ) = 2. Dr. Abdulla Eid (University of Bahrain) Logarithms 7 / 8
Notation d i E If a = 10, then we simply write log 10 as log and it is called the a l common logarithm . l u d If a = e = 2.718281828 . . . , then we simply write log e as ln and it is b called the natural logarithm . A . r D Dr. Abdulla Eid (University of Bahrain) Logarithms 8 / 8
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