Use the product, quotient, and power laws of logarithms

Alberta Grade 12 Mathematics

Relations and Functions

Expand and condense logarithmic expressions accurately

Logarithm laws provide ways to rewrite expressions while preserving their values. Multiplication inside a logarithm can become addition outside it. Division can become subtraction. An exponent on a positive argument can become a coefficient. These laws are useful for both expanding an expression into several logarithms and condensing several logarithms into one. The arguments must satisfy the conditions for the original logarithms to be defined, and the logarithms being combined must have the same base.

What you will learn

Use the product law to expand a logarithm of a product.
Use the quotient law to expand a logarithm of a quotient.
Use the power law to move an exponent between an argument and a coefficient.
Condense sums and differences of logarithms with the same base.
State the variable restrictions needed for the original logarithms to be defined.

1. The three laws

First, recall that the argument is the quantity inside a logarithm. For example, in

\log_5(3x)

, the argument is

3x

. For the logarithm to be defined in real-number work, its argument must be positive. This is the restriction we will check when variables appear.

The product law applies when the argument is a product of positive quantities. It changes one logarithm into the sum of two logarithms with the same base. For example, multiplying two positive quantities inside one logarithm corresponds to adding their logarithms.

The quotient law applies to a quotient whose numerator and denominator are positive. It changes one logarithm into the numerator's logarithm minus the denominator's logarithm. The order matters: the numerator comes first.

The power law applies when the argument being raised to a power is positive. It moves that exponent in front of the logarithm as a coefficient. In this lesson, an exponent may be an integer, including a negative integer, when the original power is defined. A negative exponent requires a nonzero base.

Each law works in reverse as well. This lets you expand a single logarithm or condense a sum or difference of logarithms. Always keep the base unchanged when applying these laws.

\log_b(MN)=\log_b M+\log_b N,\quad \log_b\left(\frac{M}{N}\right)=\log_b M-\log_b N,\quad \log_b(M^r)=r\log_b M

Product inside becomes addition outside.
Quotient inside becomes subtraction outside.
An exponent on a positive argument becomes a coefficient.
The logarithms being combined must have the same base.

2. Choosing a law and checking restrictions

Look at the structure of the argument before choosing a law. A product has factors multiplied together, so use the product law. A quotient has a numerator divided by a denominator, so use the quotient law. A power has an exponent on an argument, so use the power law.

For example,

\log_3(2(x-1))

has a product in its argument. If the original logarithm is defined, then

2(x-1)>0

, which requires

x>1

. Splitting it gives

\log_3 2+\log_3(x-1)

, and the second argument is positive under the same restriction.

A sum inside an argument is not a product. There is no product law that splits

\log_b(M+N)

into two logarithms. Likewise, a difference between two logarithms can be condensed by the quotient law, but a sum of logarithms is condensed using the product law.

Restrictions come from the original expression, not just from the final rewritten form. Check that every logarithm you write has a positive argument. If a variable occurs in a denominator, also check that the denominator is not zero. Rewriting does not make an originally undefined expression valid.

Identify multiplication, division, or an exponent before selecting a law.
Check the positivity of each logarithm argument.
Do not split a logarithm of a sum or difference.

3. A compact reference and a reliable process

The table gives both directions for each law. The left-to-right direction is useful when expanding. The right-to-left direction is useful when condensing.

For an expansion, first state the restriction, then separate products or quotients, and finally move powers to coefficients. For condensing, first move coefficients to exponents, then combine sums with the product law and differences with the quotient law.

After rewriting, compare the domain of the result with the domain of the original expression. The rewritten form should represent the original expression only where the original expression is defined.

Expand products and quotients before moving powers when that makes the structure clear.
When condensing, coefficients become exponents before logarithms are combined.
Keep domain restrictions with your final answer.

Expand and condense

Structure	Expand	Condense
Product	$\log_b(MN)=\log_b M+\log_b N$	$\log_b M+\log_b N=\log_b(MN)$
Quotient	$\log_b\left(\frac{M}{N}\right)=\log_b M-\log_b N$	$\log_b M-\log_b N=\log_b\left(\frac{M}{N}\right)$
Power	$\log_b(M^r)=r\log_b M$	$r\log_b M=\log_b(M^r)$

Worked example

Expand a product and a power

Expand

\log_5(25x^3)

using logarithm laws. State the restriction on

x

.

Check the original argument
The argument must be positive. Since 25 is positive, the condition $25x^3>0$ requires $x>0$ .
$25x^3>0\Rightarrow x>0$
Separate the product
The argument is the product of 25 and $x^3$ . The product law changes this logarithm into a sum.
$\log_5(25x^3)=\log_5 25+\log_5(x^3)$
Apply the power law
The power law moves the exponent 3 in front of the second logarithm. Since $25=5^2$ , its logarithm is 2.
$2+3\log_5 x$

Answer:

2+3\log_5 x

, for

x>0

.

Check: If

x=1

, the original expression is

\log_5 25=2

. The expanded expression is

2+3\log_5 1=2

, so the values agree.

Worked example

Condense a difference with a coefficient

Write

2\log_4(x+1)-\log_4 3

as one logarithm. State its restriction.

Find the restriction
The argument $x+1$ must be positive, so $x>-1$ . The other argument, 3, is already positive.
$x+1>0\Rightarrow x>-1$
Move the coefficient into the argument
Use the power law in reverse. The coefficient 2 becomes an exponent on the positive argument $x+1$ .
$2\log_4(x+1)=\log_4((x+1)^2)$
Combine the difference
The quotient law combines a difference of logarithms with the same base into the logarithm of a quotient.
$\log_4\left(\frac{(x+1)^2}{3}\right)$

Answer:

\log_4\left(\frac{(x+1)^2}{3}\right)

, for

x>-1

.

Check: For

x>-1

, the original argument

x+1

is positive. The condensed argument is also positive because its numerator is a positive square and its denominator is positive.

Worked example

Expand a quotient and powers

Expand

\log_2\left(\frac{a^2}{b^3}\right)

, given

a>0

and

b>0

.

Check the given restrictions
Because $a>0$ and $b>0$ , both powers are positive and the original quotient is positive. The logarithm is defined.
$a^2>0,\quad b^3>0$
Apply the quotient law
The quotient law changes the logarithm of the fraction into the logarithm of the numerator minus the logarithm of the denominator.
$\log_2(a^2)-\log_2(b^3)$
Apply the power law
Both arguments are positive, so move each exponent in front of its logarithm as a coefficient.
$2\log_2 a-3\log_2 b$

Answer:

2\log_2 a-3\log_2 b

, given

a>0

and

b>0

.

Check: For example, if

a=2

and

b=2

, the original expression is

\log_2(4/8)=-1

. The expanded expression is

2(1)-3(1)=-1

.

Common mistakes and how to avoid them

Writing

\log_b(M+N)=\log_b M+\log_b N

.

Correction: The product law applies to multiplication, not addition. Do not split a logarithm of a sum.

Combining logarithms that have different bases.

Correction: Use the product, quotient, and power laws only with logarithms of the same base.

Forgetting a positivity restriction when a variable is in an argument.

Correction: Check the original argument before rewriting, and state the restrictions that make it positive.

Changing the order when condensing a difference.

Correction: The first logarithm becomes the numerator and the second becomes the denominator.

Lesson summary

Use the product law to change multiplication inside a logarithm into addition outside.
Use the quotient law to change division inside a logarithm into subtraction outside.
Use the power law to move an exponent on a positive argument to a coefficient.
Use the same base when combining logarithms, and check the original argument restrictions.

Check your understanding

Question 1

Which expression is equivalent to

\log_3(9x)

, for

x>0

?

$2+\log_3 x$
$\log_3 9+\log_3(x^2)$
$9+\log_3 x$
$2\log_3 x$

Show answer and explanation

2+\log_3 x

The product law gives

\log_3 9+\log_3 x

. Since

9=3^2

,

\log_3 9=2

.

Question 2

Which expression is equivalent to

\log_5\left(\frac{m}{n^2}\right)

when

m>0

and

n>0

?

$\log_5 m-2\log_5 n$
$\log_5 m+2\log_5 n$
$2\log_5(m-n)$
$\frac{\log_5 m}{2\log_5 n}$

Show answer and explanation

\log_5 m-2\log_5 n

The quotient law gives

\log_5 m-\log_5(n^2)

, and the power law gives

\log_5 m-2\log_5 n

.

Question 3

Which expression is equivalent to

3\log_2 p+\log_2 q

, given

p>0

and

q>0

?

$\log_2(p^3q)$
$\log_2(3p+q)$
$\log_2\left(\frac{p^3}{q}\right)$
$\log_2(pq^3)$

Show answer and explanation

\log_2(p^3q)

The power law gives

3\log_2 p=\log_2(p^3)

. The product law then combines the sum as

\log_2(p^3q)

.

Key terms

Argument: The quantity inside a logarithm. It must be positive for the logarithm to be defined in real-number work.
Expand: Rewrite one logarithm as a sum or difference of logarithms by using the logarithm laws.
Condense: Rewrite a sum or difference of logarithms as one logarithm by using the logarithm laws.
Restriction: A condition on a variable that ensures the original expression is defined.

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About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Alberta Grade 12 Mathematics (Math 30-1), expectation RF8. It is a study resource, not an official curriculum publication.

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

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RF8 · Use the product, quotient, and power laws of logarithms

Alberta Grade 12 Mathematics

Expand and condense logarithmic expressions accurately

What you will learn

1. The three laws

2. Choosing a law and checking restrictions

3. A compact reference and a reliable process

Expand and condense

Expand a product and a power

Condense a difference with a coefficient

Expand a quotient and powers

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Key terms

Continue through Math 30-1

About this lesson and its review