Exponential and Logarithmic Functions
The Logarithmic function with base b is a function, y = logb x. Here b is greater than zero and the function x is defined for all x greater than zero. An Exponential function with base b is a function, y=bx, defined for every real number.
Inverse Function: To find an inverse function (f-1), we need to interchange x and y and then solve for y. Example: f -1( x) of 2x +1 will be, y=2x+1 (interchange x and y and solve for y)
x =2y+1
y = (x-1)/2 = f-1(x)
The Exponential functions and Logarithmic functions are inverse functions, that is, for any base b, the functions f(x) = logb x, g(x) = bx are inverses.
Example: let f(x) =ln x and g(x) = ex then f and g satisfy the inverse functions. f(g(x) = ln ex= x and g(f(x) = eln x= x, f(g(x) = g(f(x) and hence the functions f(x) and g(x) are inverses
Derivatives of Logarithmic and Exponential Functions
The most common exponential and logarithmic functions are natural exponent function ex, and the natural logarithm function, ln(x). The derivatives of Exponential and Logarithmic Functions are:
Exponential Functions derivative: d/dx (ex) = ex d/dx(ax) =ax ln a
Logarithmic Functions derivative: d/dx(ln x) =1/x d/dx(loga x)= 1/x ln a
Example: Derivative of f(x) = e3x+2 is given by d/dx (e3x+2) = e3x+2. 3 = 3e3x+2
Derivative of f(x) = ln (3x+2) is given by d/dx[ln (3x+2)] = [1/(3x+2)]. 2 = 2/(3x+2)
Graphing Exponential and Logarithmic Functions
Exponential functions play a large role in real life. From science to money, graphing these exponential functions provide a visual representations to many applications in real life. Graphs of Exponential and Logarithmic functions using examples are as follows,
Let us now graph an exponential function, f(x) = 2x. First we evaluate f(x) using the integers -3, -2, -1, 0, 1,2,3 and tabulate the values.
x -3 -2 -1 0 1 2 3
f(x) 1/8 1/4 1/2 1 2 4 8
(x,y) (-3,0.125) (-2, 0.25) (-1,0.5) (0,1) (1,2) (2,4) (3,8)
Once we get the ordered pairs (x,y) plot the points which gives us the graph of the exponential function f(x)=2x.
Graphing Logarithmic Functions: There are several ways to graph logarithmic functions. The easiest way to graph them is to re-write them in exponential form.
Example: Graph the logarithmic function, f(x) = log5 x. Re-writing f(x) = y =log5 x in exponential form we get x = 5y, choose values for y and then compute corresponding values for x. Tabulating the values,
y -1 0 1 2
x = 5^y 1/5 1 5 25
(x,y) (0.2,-1) (1,0) (5, 1) (25, 2)
Plot the (x,y) values. The graph we get is the graph of the logarithmic function, f(x)= log5 x.
