Lesson Explainer: Laws of Logarithms

In this explainer, we will learn how to use the laws of logarithms to simplify logarithmic expressions and prove logarithmic statements.

First, let’s recall what a logarithm is. A logarithmic function is the inverse function of an exponential function. We can use logarithms to find unknown exponents in exponential equations and to rewrite an exponential equation so that the exponent is the subject of the equation. We do this by using the following definition.

Definition: Logarithms

If an exponential equation is in the form , where , then it can be written as the logarithmic equation where is the base of the logarithm, is the argument, and is the exponent.

Therefore, .

Just like with exponents, we have laws for logarithms. Since logarithmic functions are the inverse of exponential functions, the laws of logarithms are the opposite of the laws of exponents. We can also use the laws of exponents to help us derive the laws of logarithms, which we will do throughout this explainer. Let’s first recall the laws of exponents.

Rule: Laws of Exponents

We will start by deriving two special cases of logarithms using the definition of a logarithm and two of the laws of exponents as follows.

Since , then setting , we can say where .

Similarly, by setting , we can say where .

This leads us to the properties below.

Property: Special Values of Logarithms

For a logarithm with base , where ,

Due to the fact that logarithms are defined to be the inverse of exponential functions and that and , the following properties hold.

Property: Composition of Exponential and Logarithmic Functions

For , we have

Also, note that is a requirement for the first property since we cannot take the logarithm of a negative number.

We will now derive each of the laws of logarithms for like bases in turn.

First, we will derive the multiplication law of logarithms, which links to the multiplication law for exponents.

Let’s say we have two logarithmic equations and , both of which have base , where , and arguments , . By rewriting these as exponents, we get and

Multiplying these two exponential equations together and using the multiplication law for exponents, we get

If we rewrite this as a logarithmic equation with base and argument , we get

Then, substituting and , we get

Therefore,

Next, we will derive the division law of logarithms that links to the division law for exponents.

Let’s say we have two logarithmic equations and , which have base , where , and arguments , . By rewriting these as exponents, we get and

Dividing the first exponential equation by the second and using the division law for exponents, we get

If we rewrite this as a logarithmic equation with base and argument , we get

Then, substituting and , we get

Therefore,

Lastly, we will derive the power law of logarithms, which links to the power law for exponents.

Let’s say we have the logarithmic equation that has base , where , and argument . By rewriting this as an exponent, we get

If we then raise both sides to the power of and use the power law for exponents, we get

If we rewrite this as a logarithmic equation with base and argument , we get

Then, substituting in the logarithmic equation, we get

Therefore,

Having derived the three logarithm laws for like bases, we summarize them in the following properties.

Property: Laws of Logarithms for Like Bases

For logarithms with the same base , where , the following rules of logarithms apply:

Multiplication law: where and .
Division law: where and .
Power law: where .

In the first example, we will determine whether a statement about logarithms is true or false using laws of logarithms with the same base.

Example 1: Using Laws of Logarithms to Verify Equality Statements

Is it true that ?

Answer

We know from the multiplication law of logarithms that

where , , and .

Since is the right-hand side of the equation, we can say

Therefore, for both sides to be equal, the arguments must be equal, meaning

Since this is not true for all values of and , the statement is false.

In the next example, we will use the laws of logarithms with the same base to compute a logarithm that would be difficult to compute without a calculator or logarithm tables.

Example 2: Computing Logarithmic Expressions Using Laws of Logarithms

Find the value of without using a calculator.

Answer

To calculate , we can use the laws of logarithms with the same base, since both terms have the same base. We have two operations in this expression, a multiplication and two subtractions. We will start by simplifying the term with the multiplication since this is a priority in the order of operations.

To simplify , we use the power law of logarithms, which states that where . Applying this to , we get

Substituting this back into the original expression, we then get

Next, we can simplify the subtractions by using the division law of logarithms, which states that where , , and .

We will start by applying this law to the first two terms, giving us

Substituting back into the expression, we get

Next, we can apply the division law again to the remaining two terms to give us

Since the base is 4, then writing the argument 64 as a power of 4, we can use the power law again.

So, this then gives us

As we know , we get

Therefore, .

When finding unknowns in logarithms, it is helpful to use the following properties for when one logarithm is equal to another.

Property: Equating Logarithms

If two logarithms with the same base are equal, then their arguments must be equal; that is, , .
If two logarithms with the same arguments are equal, then their bases must be equal; that is, , , .

In the next example, we will consider how to find unknowns in the argument of multiple logarithms by using the laws of logarithms and properties for equating logarithms.

Example 3: Finding an Unknown in a Logarithmic Equation

Find the solution set of in .

Answer

To find the solution set of , we can use the laws of logarithms since all the terms have the same base.

First, we will simplify the left-hand side of the equation by using the multiplication law of logarithms. This states that where , , and .

Therefore, applying this to the left-hand side, we get

As both sides of the equation are a single logarithm with the same base, we use the property to solve for .

Therefore, we get

Simplifying and solving for , we get

Now, since we have as the original equation and the argument of a logarithm can only be positive, then the only valid solution is .

Therefore, the solution set of is .

In the next example, we will use the laws of logarithms to find an unknown in the argument, where one term is not written as a logarithm.

Example 4: Finding an Unknown in a Logarithmic Equation

Determine the solution set of the equation in .

Answer

To find the solution set of the equation , we can use laws of logarithms to simplify this.

To use the laws of logarithms, we will start by moving all the terms with logarithms to the left-hand side of the equation by rearranging. This gives us

Next, we will use the law of multiplication of logarithms to simplify the left-hand side. The law states that where , , and .

Applying it to the left-hand side gives us

Next, to find the unknown, we can rewrite the logarithm in its exponent form, which gives us

We can now simplify and solve , which gives us

Now, since we have as the original equation and the argument must be positive, then is the only valid solution to the equation.

Therefore, the solution set of is .

As well as using laws of logarithms to simplify and evaluate expressions with multiple terms, we can use them to expand a single logarithm. In the next example, we will use the laws of logarithms with the same base to do this, but before we do, we will recall some notation about special bases.

Definition: Notation for Logarithms with Base 10

For a logarithm with base 10, we omit the base and only write log. That is, which is called the “common logarithm.”

We will use for a logarithm of base 10 in the next example.

Example 5: Expanding Logarithms

Expand using log properties.

Answer

As we are being asked to expand a logarithm using log properties, we will need to use the laws of logarithms with the same base, which are as follows:

where , , and .

In order to expand, we apply the rules in the reverse order of operations included. Since multiplication and division are of the same order, then it is usually advisable to apply the division law first as it is less likely to result in a careless error (especially if the multiplication operation is used in both the numerator and the denominator). This gives us

Next, we will apply the multiplication law to the first logarithm, giving us

Lastly, we will apply the power law to all of the terms, giving us

Therefore, .

Extension

In addition to the laws of logarithms with the same base, we have laws for changing the base of a logarithm. This is particularly helpful when there are different logarithms with different bases in an equation, as it allows us to make the bases the same throughout.

We derive the change of base rule in a similar way to the rules of logarithms with like bases, but instead of setting two logarithmic equations with the same base, we change the base and set them with the same argument. We do this as follows.

Let’s say we have two logarithmic equations and that have bases and , where , , and the same argument , where . By rewriting these as exponents, we get and

Then, by substituting into , we get

Using the law of powers of logarithms, we get

Substituting , we get

Therefore, since , we get which when dividing through by is which is the change of base rule for logarithms, as given in the rule below.

Rule: Change of Base Rule for Logarithms

For with base and argument , we write this with a chosen base using the following rule:

Note

Since we cannot divide by 0, then and (since ).

Using the change of base rule, we can derive a further rule for logarithms, which is the law of multiplicative inverse.

Let’s say that for some logarithm with base and argument , if we use the change of base rule to change the base to , then this would give us

We know from the special values of logarithms that (as the base and argument are equal); therefore,

This is made explicit in the rule below.

Rule: Law of Multiplicative Inverse for Logarithms

For with base and argument , we can swap the base and the argument using the law of multiplicative inverse, which states that

Note

Since we cannot divide by 0, then and (since ).

In the next example, we will use the change of base rule to identify which expression is equal to another.

Example 6: Using Laws of Logarithms to Match Expressions

Select the expression equal to

Answer

To determine which of the expressions is equal to , we will consider each expression in turn and apply the laws of logarithms where needed.

For option A, if then since all the logarithms in both expressions have base , we can say that the arguments in the numerator are equal and the arguments in the denominator are also equal, giving us

Therefore, this is only true when and so it is not true for all possible values of and , meaning the answer is not option A.

For option B, if

then we can see that the argument and base have been swapped for both logarithms in the numerator and denominator of the original expression to get the logarithms in the second expression. Therefore, to check if this is true, we can apply the law of multiplicative inverse for logarithms, which states that where , , and .

Applying this to the numerator of the original expression, we get and applying it to the denominator, we get

Therefore, when substituting back into the original expression, we have

As this is not option B but rather its reciprocal, this cannot be equivalent for all values of and . So, the answer is not option B.

For option C, if then we can see that the arguments of the second expression are the same as the arguments of the first expression, but the base has changed. Therefore, we can see if the two expressions are equivalent by using the change of base rule, which is as follows: where , , , and .

Applying the change of base rule to the left-hand side of the equation and changing to base , we get

Therefore, since the left-hand side is the same as the right-hand side, then .

So, option C is the correct answer.

Note that although we have found that option C is correct, we will briefly discuss why option D is not correct. Since option D is very similar to option B (but with a different argument), then by applying the same method as that used for option B, we can show that option D is false.

In the last example, we will use laws of logarithms to solve a problem in a real-life context.

Example 7: Using Laws of Logarithms to Find the Perimeter of a Given Figure

Find the perimeter of the figure below.

Answer

First, we will find an expression for the perimeter in terms of logarithms by adding the lengths of the rectangle together. This gives us the following:

To simplify this, we can use the laws of logarithms with the same base. We will start with the multiplication law, which states that where , , and .

This then gives us

Since 729 is , we can simplify the expression and evaluate it using the power law, which states that where and .

Therefore,

As , the perimeter of the rectangle is 4 m.

In this explainer, we have derived the laws of logarithms with the same base, the change of base rule, and the law of multiplicative inverse. We have used these rules to simplify and expand logarithms and solve problems involving logarithms. Let’s recap the key points.

Key Points

For logarithms with the same base , we have the following laws:
- Multiplication law: where and .
- Division law: where and .
- Power law: where .
We have two special cases of logarithms, which are
- , ;
- , .
For solving logarithms that are equal, we can use the following properties:
- , ;
- , , .
Extension: We can change the base of a logarithm using the following laws:
- Change of base rule: where , , , and .
- Multiplicative inverse: where , , and .