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 y=b^x for 0 < b < 1. Since y=b^x is a one-to-one function, its reflection in the line y = x is also a function. The function x=b^y is the inverse function of y=b^x.

The equation of a function is usually solved for y in terms of x. To solve the equation x=b^y for y, we need to introduce some new terminology. First we will describe y in words:

Also Read:

x=b^y : “y is the exponent to the base b such that the power is x.”

A logarithm is an exponent. Therefore, we can write:

x=b^y : “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=\log_b\text{ x} : “y is the logarithm to the base b of x.”

The base b is written as a subscript to the word “log.”

x=b^y can be written as y=\log_b\text{ x}.

For example, let b = 2. Write pairs of values for x=2^y and y=\log_2x

We say that y=\log_bx, with b a positive number not equal to 1, is a logarithmic function.

Also Read : Logarithmic form of an exponential equation

Example 1:

Write the equation x=10^y for y in terms of x.

Solution:

x=10^y

y is the exponent or logarithm to the base 10 of x.

y=\log_{10}x.

Graphs of Logarithmic Functions

From our study of exponential functions, we know that when b > 1 and when 0 < b < 1, y=b^x is defined for all real values of x. Therefore, the domain of y=b^x 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 y=b^x is the set of positive real numbers.

When we interchange x and y to form the inverse function x=b^y or y=\log_bx ;

  • The domain of y=\log_bx is the set of positive real numbers.
  • The range y=\log_bx is the set of real numbers.
  • The y-axis or the line x = 0 is a vertical asymptote of y=\log_bx.

Example 2:

Sketch the graph of f(x)=2^x and Write the equation of f-1(x) and sketch the graph.

Solution:

Make a table of values for f(x)=2^x, plot the points, and draw the curve.

Let f(x)=2^x or y=2^x.

To write f^{-1}(x), interchange x and y.

x=2^y is written as y=\log_2x. therefore, f^{-1}(x)=\log_2x.

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).

 

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