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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.
Inverse Operations: 1) Addition and Subtraction 2) Multiplication and Division 3) Power and Root
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?
inverse of 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

Here, is called the base in both the logarithm and the exponential expression. Logarithms are defined only for positive values of and where is not equal to To see the implications of this definition, a particular example will be considered.
In this equation, the definition of logarithm implies that is the exponent to which the base must be raised in order to obtain
The following diagram illustrates how a logarithmic form has an equivalent exponential form using example numerical values.
Equivalence between logarithmic and exponential expressions

Why

Logarithms are only defined for positive values
Logarithms 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.

Paulina thinking of the definition of a logarithm

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
b Rewrite the logarithmic equation as an exponential equation.
c Rewrite the exponential equation as a logarithmic equation.

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.
With this definition in mind, let be the value of
Given that is the base, Paulina should ask herself what number it must be raised to in order to reach Well, to the power of is equal to the value of is Therefore,
b Similar to Part A, to rewrite as an exponential equation, the definition of a logarithm will be used.
Therefore, Paulina should substitute and in the above definition.
c To rewrite as a logarithmic equation, the definition of a logarithm will be used one last time.
Here, and will be substituted into the definition.
Pop Quiz

Evaluating Logarithms

Evaluate the logarithms.

evaluate the expression
Pop Quiz

Rewriting Logarithmic and Exponential Equations

Rewrite the logarithmic equations as exponential equations and the exponential equations as logarithmic equations.

rewrite the equation
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.

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.

This property is only valid for positive values of and and for As an example, the expression can be rewritten using this property.

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.

This property is valid for positive values of and and for For example, the expression can be rewritten using this property.

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.

This property is valid for positive values of and and for For example, can be rewritten using this property.

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.
Paulina thinking about the properties of logarithms
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
b
c

Solution

a Consider the Product Property of Logarithms.
Knowing that and that this property can be used to evaluate the given expression. Start by rewriting as the product of and

,

b Because of the quotient inside the logarithm, the Quotient Property of Logarithms will be used this time.
Also, as a consequence of the definition of a logarithm, it is known that for any positive number different than Using this information, the above property, and the fact that the given expression can be evaluated.

c Recall the Power Property of Logarithms and the Product Property of Logarithms.
Also, as a consequence of the definition of a logarithm, it is known that for any positive number different than Using this information, the above properties, and the fact that the given expression can be evaluated.

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!
Paulina thinking about using the properties of logarithms to expand expressions that involve logarithms
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
b

Hint

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.
Here, and are positive, where The main operation in the given logarithmic expression is a division. Therefore, the first property that should be used is the Quotient Property of Logarithms. Then, the Product Property of Logarithms and the Power Property of Logarithms will be applied.

Calculate logarithm

b To use the Power Property of Logarithms, the square root will be written as a rational exponent.
Now, the mentioned property can be used. Then, the Quotient Property of Logarithms and the Product Property of Logarithms can also be used.

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.
Paulina thinking about using the properties of logarithms to condense expressions that involve logarithms
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
b

Hint

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.
These three properties will be used to condense the expression.

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.
inverse of logarithm is raising to the power of x, and inverse of raising to the power of x is a logarithm
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.