Math RE - Calculus I Exponential & Logarithmic Functions Page 1 of 9. y = f(x) = 2 x. y = f(x)

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1 Math 20-0-RE - Calculus I Eponential & Logarithmic Functions Page of 9 Eponential Function The general form of the eponential function equation is = f) = a where a is a real number called the base of the eponential function and the eponent of the base is the variable. The base a cannot equal and is also a positive number greater than 0. Observe the net graphs, the behavior of each function is that as value increases, value increases rapidl. Graph of the Eponential Function = f) = 2 = f)

2 Math 20-0-RE - Calculus I Eponential & Logarithmic Functions Page 2 of 9 Graph 2 of the Eponential Function = f) = 5 = f) Graph of the Eponential Function = f) = ) = = f)

3 Math 20-0-RE - Calculus I Eponential & Logarithmic Functions Page of 9 Properties of Eponential Function ) a 0 = 2) a = a ) a n a m = an m 4) a m.a n = a m+n 5) a m ) p = a m. p 6) a + = + 7) a = 0 8) a n = a n Common bases for Eponentials The most common base is e an irrational number like π and the decimal value is e the calculator has 2 kes for 2 common bases: base 0 has ke 0 and base e has ke e. The graph of = e is shown below

4 Math 20-0-RE - Calculus I Eponential & Logarithmic Functions Page 4 of 9 Eamples for Eponentials a) Reduce: e 4.e 5.e /2 = e 4 5+/2 = e /2 b) Simplif = ) = 4 7 c) Epand: e 2 + e ) 2 = e 2 ) 2 + 2e 2 e + e ) 2 = e 4 + 2e + e 2 d) Multipl: + ). 9 + ) = = Inverse Functions The inverse operation consists of interchanging variables; therefore interchanging domain and range. If we appl the inverse operation to the eponential function = a, we get = a and this equation is called the eponential notation of the logarithmic function. The domain of the eponential function is all reals and the range of the eponential function is greater than 0. Inversel, the domain of the logarithmic function is greater than 0 and the range of the logarithmic function is all reals. Note: the behavior of the eponential function is that as value increases, value increases rapidl. Inversel, as value increases the value increases slowl for the logarithmic function.

5 Math 20-0-RE - Calculus I Eponential & Logarithmic Functions Page 5 of 9 Logarithmic Function: The general form of the logarithmic function equation has 2 shapes: eponential notation and the logarithmic notation as shown below: = a = log a ) where a is a real number called the base of the logarithmic function. The base a cannot equal and is also a positive number greater than 0. Observe the net graphs, the behavior of each function is that as value increases, value increases slowl. Graph of the Logarithmic Function = 2 = log 2 ) = f) = log 2 )

6 Math 20-0-RE - Calculus I Eponential & Logarithmic Functions Page 6 of 9 Graph 2 of the Logarithmic Function = 5 = log 5 ) = f) = log 5 ) Graph of the Logarithmic Function = = log ) = f) = log )

7 Math 20-0-RE - Calculus I Eponential & Logarithmic Functions Page 7 of 9 Properties of Logarithmic Function ) log a ) = 0 2) log a 0 + ) = m ) ) log a = log n a m) log a n) 4) log a m.n) = log a m) + log a n) 5) log a m p ) = p log a m) 6) log a + ) = + 7) log ) ) a m = log a = log m a m) Common bases for Logarithms The most common base is e an irrational number like π and the decimal value is e the calculator has 2 kes for 2 common bases: base 0 has ke log) and base e has ke ln). The graph of = ln) is shown below

8 Math 20-0-RE - Calculus I Eponential & Logarithmic Functions Page 8 of 9 Eamples for Logarithms a) Reduce: log + ) + log ) = log 2 ) ) b) Simplif log 5 ) log 5 ) = log 5 ) c) Epand into simple logs: log 6 = log ) log 6 + 2) = log 6 ) log 6 + 2) d) Simplif: 2 log 8 ) log ) log 8 2 ) log 8 [ 2 + ) ] = log 8 2 ) log 8 2 ) log 8 + ) = log 8 + )) Properties of Eponentials & Logarithms ) log a a ) = log e e ) = ln e ) = 2) a log a ) = e log e ) = e ln) = Eamples: ) 0 log 0 2) = 0 log 2) = 2 2) log e e 4 ) = ln e 4) = 4 ) Simplif: log 4 4 2) = 2 4) Simplif: 5 log 5 ) = 5 log 5 ) = 5) Simplif: e ln2 ) = e ln 2 ) = 2 = 2 6) Simplif: 2 ln e 2) = ln e 2) 2 = ln e 22) = 2 2

9 Math 20-0-RE - Calculus I Eponential & Logarithmic Functions Page 9 of 9 Graph of Eponential & Logarithmic Functions The graph below shows the eponential function brown curve) = 2 and its inverse equation the logarithmic function blue curve) = 2 which is the same as = log 2 ). Notice the smmetr of both curves with the red) line =. Also point,2) on the eponential function has its smmetrical point 2, ) on logarithmic function. The domain of eponential function is all reals becomes the range of logarithmic function; also the range of eponential function is greater than 0 becomes the domain of logarithmic function. = 2 2 = = log 2 )

Intermediate Algebra Section 9.3 Logarithmic Functions

Intermediate Algebra Section 9.3 Logarithmic Functions We have studied inverse functions, learning when they eist and how to find them. If we look at the graph of the eponential function, f ( ) = a, where