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The Relationship Between Exponential and Logarithmic Functions

Author: Sophia

what's covered

In this lesson, you will use exponential functions to define and establish relationships with logarithmic functions. Specifically, this lesson will cover:

Inverses of Exponential Functions and Logarithms
Writing Logarithmic Equations in Exponential Form
Writing Exponential Equations in Logarithmic Form

1. Inverses of Exponential Functions and Logarithms

Consider the function f open parentheses x close parentheses equals 2 to the power of x comma whose table of values and graph are shown below.


0
1
2
3
4

Domain:
Range:
Horizontal asymptote:

Since the graph of f open parentheses x close parentheses equals b to the power of x

passes the horizontal line test, f open parentheses x close parentheses

is one-to-one and therefore has an inverse function. We say that f open parentheses x close parentheses

has the one-to-one property.

The one-to-one property for exponential functions tells us that if two exponential expressions with the same base are equal to each other, then the exponents must also be equal. The property also tells us that if two quantities are equal, then using them both as exponents on the same base will produce the same result.

This property will be used more frequently when solving equations.

Recall that to find an inverse of a one-to-one function, interchange its inputs and outputs. The table of values of the inverse function are shown below.


	0
	1
	2
	3
	4

Notice some patterns in the inverse function. For example:

f to the power of short dash 1 end exponent open parentheses 16 close parentheses equals f to the power of short dash 1 end exponent open parentheses 2 to the power of 4 close parentheses equals 4

f to the power of short dash 1 end exponent open parentheses 1 close parentheses equals f to the power of short dash 1 end exponent open parentheses 2 to the power of 0 close parentheses equals 0

The output of the inverse function is the exponent of the input (when the base is 2).

Since the inverse of an exponential function is important, we need to give it a name.

		Replace with y.
		Interchange x and y.

Another word for exponent is logarithm.

We rewrite the equation x equals 2 to the power of y as y equals log subscript 2 x comma which is “the logarithm with base 2 of x.”

Therefore, the inverse of f open parentheses x close parentheses equals 2 to the power of x is f to the power of short dash 1 end exponent open parentheses x close parentheses equals log subscript 2 x.

big idea

For a base b greater than 0

and

the inverse of f open parentheses x close parentheses equals b to the power of x

f to the power of short dash 1 end exponent open parentheses x close parentheses equals log subscript b x.

A logarithm can be written as log subscript b x

log subscript b open parentheses x close parentheses.

The quantity x is often called the argument of the function.

Naturally, if the argument is a more complex expression, then parentheses are required. For base 10,
“ log subscript 10 x

” is written as “ log x

.” This means y equals log x

is equivalent to 10 to the power of y equals x.

Note: a logarithm with base 10 is also called a common logarithm.

For base e, “ log subscript e x

” is written as “ ln x

,” which is called the “natural logarithm of x.” This means y equals ln x

is equivalent to e to the power of y equals x.

did you know

The reason that we write “ log x

” in place of “ log subscript 10 x

” is because our number system is base 10 (which comes from us having 10 fingers). It is also easier to measure in powers of 10. That said, there is nothing incorrect about writing “ log subscript 10 x

,” it is just that “ log x

” is a shorter way of writing it.

EXAMPLE

Consider the functions f open parentheses x close parentheses equals 10 to the power of x

and

g open parentheses x close parentheses equals e to the power of x.

Then,

f to the power of short dash 1 end exponent open parentheses x close parentheses equals log subscript 10 x comma

which is then written f to the power of short dash 1 end exponent open parentheses x close parentheses equals log x.

Also,

g to the power of short dash 1 end exponent open parentheses x close parentheses equals log subscript e x comma

which is written g to the power of short dash 1 end exponent open parentheses x close parentheses equals ln x.

try it

Consider the function f open parentheses x close parentheses equals 4 to the power of x.

Find the inverse of f (x ).

f to the power of short dash 1 end exponent open parentheses x close parentheses equals log subscript 4 x

Using the definition of a logarithm is useful for writing a logarithmic expression in exponential form, and vice versa. First, we will look at writing a logarithm in exponential form.

terms to know

One-To-One Property for Exponential Functions

Consider the function f open parentheses x close parentheses equals b to the power of x comma

where

and

Then, for quantities R and S, b to the power of R equals b to the power of S

if and only if R equals S.

Logarithm

Written as log subscript b x comma

it is the real number y for which b to the power of y equals x comma

where

, and

Common Logarithm

A logarithm whose base is 10. It is written log x

in place of log subscript 10 x.

then

Natural Logarithm

A logarithm whose base is e. It is written ln x

in place of log subscript e x.

then

2. Writing Logarithmic Equations in Exponential Form

Given a logarithmic equation, we can use the fact that y equals log subscript b x means x equals b to the power of y to write the corresponding exponential equation.

Where appropriate, remember that log subscript e x is written as ln x , and log subscript 10 x is written as log x.

EXAMPLE

Convert each logarithmic equation to its corresponding exponential form.

Logarithmic Equation	Exponential Form
	This equation in exponential form is
	Remember that “log 0.01” with no base written is assumed to be base 10. If it helps, write as Then, the exponential form is
	This equation in exponential form is
	Remember that “” means that the base is e. If it helps you, write as Then, the exponential form is

try it

Consider the logarithmic equation 2 equals log subscript 3 x.

Write the logarithmic equation in exponential form.

try it

Consider the logarithmic equation x equals ln 4.

Write the logarithmic equation in exponential form.

try it

Consider the logarithmic equation log 1000 equals 3.

Write the logarithmic equation in exponential form.

In some circumstances, notice that writing the equation in exponential form helps to see what the value of x is that makes the statement true. For example, the logarithmic equation 2 equals log subscript 3 x has exponential form 3 squared equals x comma which means x equals 9. In this case, writing the exponential form helped to solve the equation for x.

Now, we’ll write exponential equations in their corresponding logarithmic form.

3. Writing Exponential Equations in Logarithmic Form

From earlier, we know that given b to the power of y equals x comma then log subscript b x equals y. Remember the special cases for base 10 and base e.

EXAMPLE

Convert each exponential equation to its corresponding logarithmic form.

Exponential Form	Logarithmic Equation
	Its corresponding logarithmic form is
	The corresponding logarithmic form is To simplify this, remember that “” can be replaced with “.” The final answer is
	The corresponding logarithmic form is To simplify this, remember that “” can be replaced with “.” The final answer is

try it

Consider the equation 3 to the power of short dash 2 end exponent equals 1 over 9.

Write the equation in logarithmic form.

log subscript 3 open parentheses 1 over 9 close parentheses equals short dash 2

try it

Consider the equation e to the power of short dash 4 t end exponent equals y.

Write the equation in logarithmic form.

try it

Consider the equation 10 to the power of 3 divided by 2 end exponent equals x.

Write the equation in logarithmic form.

summary

In this lesson, you learned that a logarithmic function is the inverse of an exponential function with the same base. Because of the relationship between expressions in logarithmic form and exponential form, you can convert equations between logarithmic form and exponential form. You learned that when writing logarithmic equations in exponential form, you can use the fact that y equals log subscript b x

means

to write the corresponding exponential form. Conversely, when writing exponential equations in logarithmic form, you learned that given b to the power of y equals x comma

then

, noting the special cases for base 10 and base e.

SOURCE: THIS WORK IS ADAPTED FROM PRECALCULUS BY JAY ABRAMSON. ACCESS FOR FREE AT OPENSTAX.ORG/BOOKS/PRECALCULUS/PAGES/1-INTRODUCTION-TO-FUNCTIONS

Terms to Know

Common Logarithm

A logarithm whose base is 10. It is written $log x$ in place of $log subscript 10 x.$

If $y equals log x comma$ then $x equals 10 to the power of y.$

Logarithm

Written as $log subscript b x comma$ it is the real number y for which $b to the power of y equals x comma$ where $x greater than 0 comma$ $b greater than 0$ , and $b not equal to 1.$

Natural Logarithm

A logarithm whose base is e. It is written $ln x$ in place of $log subscript e x.$

If $y equals ln x comma$ then $x equals e to the power of y.$

One-To-One Property for Exponential Functions

Consider the function $f open parentheses x close parentheses equals b to the power of x comma$ where $b greater than 0$ and $b not equal to 1.$ Then, for quantities R and S, $b to the power of R equals b to the power of S$ if and only if $R equals S.$