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Inverse operations are mathematical operations that, in a way, undo each other. This lesson will discuss logarithims, which are the inverse operation of raising a number to a variable. Properties of logarithms will also be described.
Catch-Up and Review
Here are a few recommended readings before getting started with this lesson.
Explore
What Is the Inverse Operation of Calculating an Exponent?
Sometimes the inverse of a mathematical operation is clearly identifiable. For example, addition and subtraction, multiplication and division, or raising to the power of and calculating the root are inverse operations.
However, in some other cases, inverse operations are a bit more difficult to come up with. What is an operation that undoes calculating an exponent? 
Discussion
Logarithm
A logarithm is the inverse function of an exponential function. The logarithm of a positive number is written as and read as the logarithm of with base
Why
Logarithms are only defined for positive valuesLogarithms are undefined for non-positive values. The reason behind this can be explained by rewriting the logarithm in exponential form.
Because is a positive number, the expression is always positive. Therefore, since the value of is always positive. This means that can never be negative. In this case, is the expression in the logarithm. Therefore, expressions in logarithmic functions must be positive.
As a consequence of the definition of a logarithm, two properties can be deduced. In these properties, is positive and not equal to
| Property | Reason |
|---|---|
| A number raised to the power of is equal to itself. | |
| A number raised to the power of is equal to |
Example
Evaluating and Rewriting Logarithmic Expressions
Paulina has recently become excited learning about logarithms.
She eagerly went to her math teacher and asked for some introductory exercises to practice evaluating and rewriting logarithmic expressions. Help her get off to a good start!
a Evaluate the expression
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":["x"],"constants":["PI"]},"rendered":false},"text":""},"formTextBefore":null,"formTextAfter":null,"answer":{"text":["3"]}}
b Rewrite the logarithmic equation as an exponential equation.
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":["x"],"constants":["a"]},"rendered":false},"text":""},"formTextBefore":null,"formTextAfter":null,"answer":{"text":["a=4^x","4^x=a"]}}
c Rewrite the exponential equation as a logarithmic equation.
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":["x"],"constants":["a"]},"rendered":false},"text":""},"formTextBefore":null,"formTextAfter":null,"answer":{"text":["x=\\log_3a","\\log_3a=x","x=\\log_3(a)","\\log_3(a)=x"]}}
Hint
a To what power must the base of be raised to obtain
b Use the definition of a logarithm.
c How can an exponential equation be rewritten using a logarithm?
Solution
a To evaluate the given expression, it is helpful to recall what the definition of a logarithm is.
b Similar to Part A, to rewrite as an exponential equation, the definition of a logarithm will be used.
c To rewrite as a logarithmic equation, the definition of a logarithm will be used one last time.
Pop Quiz
Evaluating Logarithms
Evaluate the logarithms.
Pop Quiz
Rewriting Logarithmic and Exponential Equations
Rewrite the logarithmic equations as exponential equations and the exponential equations as logarithmic equations.
Discussion
Properties of Logarithms
For the proof of these properties, two identities will be used. Start by recalling the definition of a logarithm.
The first equation of the definition states that Therefore can be substituted for in the second equation. Furthermore, the second equation states that is equal to This means that can be substituted for in the first equation.
This property is only valid for positive values of and and for As an example, the expression can be rewritten using this property.
The obtained identities will be used together with the Product of Powers Property to prove the Product Property of Logarithms.
This property is valid for positive values of and and for For example, the expression can be rewritten using this property.
The obtained identities will be used together with the Quotient of Powers Property to prove the Quotient Property of Logarithms.
This property is valid for positive values of and and for For example, can be rewritten using this property.
The obtained identities will be used together with the Power of a Power Property to prove the Power Property of Logarithms.
With this information in mind, three properties can be stated.
Rule
Product Property of Logarithms
The logarithm of a product can be written as the sum of the individual logarithms of each factor.
Proof
The obtained identities will be used together with the Product of Powers Property to prove the Product Property of Logarithms.
Rule
Quotient Property of Logarithms
The logarithm of a quotient can be written as the difference between the logarithm of the numerator and the logarithm of the denominator.
Proof
The obtained identities will be used together with the Quotient of Powers Property to prove the Quotient Property of Logarithms.
Rule
Power Property of Logarithms
The logarithm of a power can be written as the product of the exponent and the logarithm of the base.
Proof
The obtained identities will be used together with the Power of a Power Property to prove the Power Property of Logarithms.
Example
Using Properties to Evaluate Expressions
After understanding the definition of a logarithm and learning about its properties, Paulina is ready to delve deeper into this topic.
As a part of her deep dive into properties of logarithms, she begins with two approximations.
Without using a calculator, only relying on the two given approximations, Paulina wants to evaluate three logarithmic expressions. This will make her feel like she is really understaning how to operate with logarithms. Help her find the answer! Write each answer rounded to three decimal places.
a
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b
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":["x"],"constants":["PI"]},"rendered":false},"text":""},"formTextBefore":null,"formTextAfter":null,"answer":{"text":["- 2.322"]}}
c
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":["x"],"constants":["PI"]},"rendered":false},"text":""},"formTextBefore":null,"formTextAfter":null,"answer":{"text":["4.585"]}}
Hint
a Use the Product Property of Logarithms.
b Use the Quotient Property of Logarithms.
c Use the Power Property of Logarithms.
Solution
a Consider the Product Property of Logarithms.
b Because of the quotient inside the logarithm, the Quotient Property of Logarithms will be used this time.
c Recall the Power Property of Logarithms and the Product Property of Logarithms.
SplitIntoFactors
Split into factors
WritePow
Write as a power
IdPropMult
Identity Property of Multiplication
AddTerms
Add terms
Example
Using Properties to Expand Expressions
Paulina has moved beyond only understanding the definition of a logarithm. She can now rewrite exponential equations as logarithmic equations and convert logarithmic equations into exponential equations. On top of all of that, she can even evaluate logarithmic expressions. Paulina is starting to master this topic! 
Now, there are even more interesting challenges to face. This time, Paulina needs to expand two logarithmic expressions by using the Properties of Logarithms. Help her do this! Assume that all the variables involved are positive.
a
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":["x","y"],"constants":[]},"rendered":false},"text":""},"formTextBefore":null,"formTextAfter":null,"answer":{"text":["2+3\\log_5 x-2\\log_5 y","2-2\\log_5 y+3\\log_5 x","3\\log_5 x-2\\log_5 y+2","3\\log_5 x+2-2\\log_5 y","-2\\log_5 y+2+3\\log_5 x","-2\\log_5 y+3\\log_5 x+2"]}}
b
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":["x","y"],"constants":[]},"rendered":false},"text":""},"formTextBefore":null,"formTextAfter":null,"answer":{"text":["\\dfrac{1}{2} \\log_4 x+\\dfrac{1}{2} \\log_4 y-\\dfrac{1}{2}","\\dfrac{1}{2} \\log_4 x-\\dfrac{1}{2}+\\dfrac{1}{2} \\log_4 y","\\dfrac{1}{2} \\log_4 y+\\dfrac{1}{2} \\log_4 x-\\dfrac{1}{2}","\\dfrac{1}{2} \\log_4 y-\\dfrac{1}{2}+\\dfrac{1}{2} \\log_4 x","-\\dfrac{1}{2}+\\dfrac{1}{2} \\log_4 x+\\dfrac{1}{2} \\log_4 y","-\\dfrac{1}{2}+\\dfrac{1}{2} \\log_4 y+\\dfrac{1}{2} \\log_4 x"]}}
Hint
a Start by using the Quotient Property of Logarithms. Then, use the Product Property of Logarithms. Finally, use the Power Property of Logarithms.
b Start by rewriting the square root as a rational exponent. Then, use the Power Property of Logarithms to remove the exponent.
Solution
a Consider the Properties of Logarithms.
Calculate logarithm
b To use the Power Property of Logarithms, the square root will be written as a rational exponent.
Example
Using Properties to Condense Expressions
Paulina feels extremely confident about her skills in using logarithms! Now, instead of expanding algebraic and numeric expressions that involve logarithms, she will practice condensing them.
To finally master logarithmic expressions with and without variables, Paulina wants to condense two expressions using the Properties of Logarithms. Help her do this! Assume that all variables involved are positive.
a
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":[],"constants":[]},"rendered":false},"text":""},"formTextBefore":null,"formTextAfter":null,"answer":{"text":["\\log_2 250"]}}
b
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":["x","y"],"constants":[]},"rendered":false},"text":""},"formTextBefore":null,"formTextAfter":null,"answer":{"text":["\\log_3 \\dfrac{32y^2}{x^6}","\\log_3 (\\dfrac{32y^2}{x^6})"]}}
Hint
a Use the Product Property of Logarithms, the Power Property of Logarithms, and the Quotient Property of Logarithms.
b Use the Properties of Logarithms.
Solution
a To condense this expression, three properties of logarithms will be recalled first. These are the Product Property of Logarithms, the Quotient Property of Logarithms, and the Power Property of Logarithms.
b Similar to Part A, to condense this expression, the same three Properties of Logarithms will be applied.
Closure
Inverse Operations
In this lesson, it has been presented that exponents and logarithms are their own inverses. This means that these two operations essentially undo each other.
The definition of a logarithm and important Properties of Logarithms have also been taught and practiced in this lesson.
| Definition | |
|---|---|
| Identity Derived From the Definition | |
| Identity Derived From the Definition | |
| Product Property of Logarithms | |
| Quotient Property of Logarithms | |
| Power Property of Logarithms |
It is important to keep in mind that these properties are only valid for positive values of and where Furthermore, these properties can be used in several situations.
- Writing a power as a logarithm and writing a logarithm as a power.
- Evaluating logarithms.
- Using given approximations of certain logarithms to approximate other logarithmic expressions.
- Expanding and condensing logarithmic expressions with and without variables.
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