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The Relationship Between Exponential and Logarithmic Functions

Author: Sophia
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what's covered
In this lesson, you will use exponential functions to define and establish relationships with logarithmic functions. Specifically, this lesson will cover:

Table of Contents

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.

bold italic x bold italic f open parentheses bold x close parentheses
0 2 to the power of 0 equals 1
1 2 to the power of 1 equals 2
2 2 squared equals 4
3 2 cubed equals 8
4 2 to the power of 4 equals 16 









    

  • Domain: open parentheses short dash infinity comma space infinity close parentheses

    • 

  • Range: open parentheses 0 comma space infinity close parentheses

    • 

  • Horizontal asymptote: y equals 0

    • 


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. 







bold italic x


bold italic f to the power of bold short dash bold 1 end exponent open parentheses bold x close parentheses






2 to the power of 0 equals 1


0






2 to the power of 1 equals 2


1






2 squared equals 4


2






2 cubed equals 8


3






2 to the power of 4 equals 16


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. 







y equals 2 to the power of x


Replace f open parentheses x close parentheses with y.






x equals 2 to the power of 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 b not equal to 1 comma the inverse of f open parentheses x close parentheses equals b to the power of x is 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 or 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 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.



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.



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. 


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.















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






y equals log subscript 4 x


This equation in exponential form is 4 to the power of y equals x.






log     0.01 equals short dash 2


Remember that “log 0.01” with no base written is assumed to be base 10. If it helps, write log     0.01 equals short dash 2 as log subscript 10 0.01 equals short dash 2.



Then, the exponential form is 10 to the power of short dash 2 end exponent equals 0.01.






log subscript 2 32 equals 5


This equation in exponential form is 2 to the power of 5 equals 32.






ln     y equals 5


Remember that “ln     y” means that the base is e. If it helps you, write ln     y equals 5 as log subscript e y equals 5.



Then, the exponential form is e to the power of 5 equals y.

























try it





Consider the logarithmic equation 2 equals log subscript 3 x. 




Write the logarithmic equation in exponential form.
3 squared equals x

























try it





Consider the logarithmic equation x equals ln     4. 




Write the logarithmic equation in exponential form.
Recall that “ln” corresponds to base “e.”  



It could help to rewrite the equation as x equals log subscript e 4. Then, the corresponding exponential equation is e to the power of x equals 4.


























try it





Consider the logarithmic equation log     1000 equals 3.




Write the logarithmic equation in exponential form.
Recall that “log” corresponds to base “10.”  



It could help to rewrite the equation as log subscript 10 1000 equals 3. Then, the corresponding exponential equation is 10 cubed equals 1000.













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






4 to the power of 0.5 end exponent equals 2


Its corresponding logarithmic form is log subscript 4 2 equals 0.5.






e to the power of 0 equals 1


The corresponding logarithmic form is log subscript e 1 equals 0. To simplify this, remember that “log subscript e 1” can be replaced with “ln     1.” 



The final answer is ln     1 equals 0.






10 squared equals 100


The corresponding logarithmic form is log subscript 10 100 equals 2. To simplify this, remember that “log subscript 10 100” can be replaced with “log     100.” 



The final answer is log     100 equals 2.

























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.
Use the definition of logarithm to rewrite: log subscript e y equals short dash 4 t



Recall that we use the “ln” notation to represent logarithms with base “e.” It is proper to rewrite the equation as ln     y equals short dash 4 t comma which is the final answer.

























try it





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




Write the equation in logarithmic form.
Use the definition of logarithm to rewrite: log subscript 10 x equals 3 over 2



Recall that we use the “log” notation to represent logarithms with base 10. It is proper to rewrite the equation as log     x equals 3 over 2 comma which is the final answer.






















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 x equals b to the power of y 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 log subscript b x equals y, noting the special cases for base 10 and base e.








SOURCE: THIS TUTORIAL HAS BEEN ADAPTED FROM OPENSTAX "PRECALCULUS” BY JAY ABRAMSON. ACCESS FOR FREE AT OPENSTAX.ORG/DETAILS/BOOKS/PRECALCULUS-2E. LICENSE: CREATIVE COMMONS ATTRIBUTION 4.0 INTERNATIONAL.



















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.
































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