# 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:

$x=b^y$ : “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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