Inverse of an exponential function and Graphs of Logarithmic Functions
Any positive real number can be the exponent of a power by drawing the graph of the exponential function for 0 < b < 1. Since is a one-to-one function, its reflection in the line y = x is also a function. The function is the inverse function of .
The equation of a function is usually solved for y in terms of x. To solve the equation for y, we need to introduce some new terminology. First we will describe y in words:
: “y is the exponent to the base b such that the power is x.”
A logarithm is an exponent. Therefore, we can write:
: “y is the logarithm to the base b of the power x.”
The word logarithm is abbreviated as log. Look at the essential parts of this sentence:
: “y is the logarithm to the base b of x.”
The base b is written as a subscript to the word “log.”
can be written as .
For example, let b = 2. Write pairs of values for and
We say that , with b a positive number not equal to 1, is a logarithmic function.
Also Read : Logarithmic form of an exponential equation
Write the equation for y in terms of x.
y is the exponent or logarithm to the base 10 of x.
Graphs of Logarithmic Functions
From our study of exponential functions, we know that when b > 1 and when 0 < b < 1, is defined for all real values of x. Therefore, the domain of is the set of real numbers. When b > 1, as the negative values of x get larger and larger in absolute value, the value of b gets smaller but is always positive. When 0 < b < 1, as the positive values of x get larger and larger, the value of bx gets smaller but is always positive. Therefore, the range of is the set of positive real numbers.
When we interchange x and y to form the inverse function or ;
- The domain of is the set of positive real numbers.
- The range is the set of real numbers.
- The y-axis or the line x = 0 is a vertical asymptote of .
Sketch the graph of and Write the equation of f-1(x) and sketch the graph.
Make a table of values for , plot the points, and draw the curve.
Let or .
To write , interchange x and y.
is written as . therefore, .
To draw the graph, interchange x and y in each ordered pair or reflect the graph of f(x) over the line y = x. ordered pairs of f-1(x) include ( 1/4 , -2), ( ½ , -1), (1,0) , (2,1), (4, 2), and (8, 3).