# 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:

Also Read:

: “*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

**log**arithm to the

**base**of

*b***.”**

*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**

Example 1:

Write the equation for *y *in terms of *x*.

Solution:

*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 *b ^{x} *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

Example 2:

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

Solution:

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).