Apply the laws of logarithms and exponents

Ontario Grade 12 Mathematics

Exponential and Logarithmic Functions

Grade 12 Mathematics — A1.4

Exponent and logarithm laws help you rewrite expressions without changing their values. You can use them to simplify calculations, combine repeated factors, or solve equations. This lesson reviews the exponent rules that logarithms depend on, then builds the logarithm rules from the meaning of a logarithm. In every logarithm, the base must be positive and different from

1

, and the argument must be positive.

What you will learn

Use exponent laws to rewrite and simplify expressions.
Connect logarithms to exponential form.
Apply logarithm laws to expand or combine logarithmic expressions.
Check restrictions and verify a result.

1. Prerequisite bridge: what exponent laws say

An exponent tells how many times a base is used as a factor. For example,

3^4

means

3\times3\times3\times3

. Repeated factors can be grouped, which explains why powers with the same base follow useful rules.

When multiplying powers with the same base, combine the repeated factors by adding their exponents. When dividing, cancel matching factors, so subtract the exponents. A power raised to another power multiplies the exponents. These rules require the same base for multiplication and division.

A zero exponent gives

1

for any nonzero base. A negative exponent means take the reciprocal. A fractional exponent represents a root: for example, an exponent of

\frac{1}{2}

means square root. These forms let you express and simplify powers in different ways.

a^m a^n=a^{m+n},\quad \frac{a^m}{a^n}=a^{m-n},\quad (a^m)^n=a^{mn}

For equal bases, multiplication adds exponents and division subtracts them.
A power of a power multiplies exponents.
A negative exponent indicates a reciprocal; a zero exponent gives $1$ when the base is nonzero.

2. Logarithms reverse exponentiation

A logarithm answers an exponent question: what exponent on a base produces a given number? The statement

\log_b x=y

means that

b

raised to

y

equals

x

. This is the link between logarithms and exponents, not a separate calculation rule.

For example, since

2^5=32

, it follows that

\log_2 32=5

. In a logarithm,

b

is the base and

x

is the argument, the number inside the logarithm. The base must satisfy

b>0

and

b\ne1

. The argument must satisfy

x>0

.

The logarithm of a product can be split into a sum. The logarithm of a quotient can be split into a difference. A power in the argument can be moved in front as a factor. These rules work because the corresponding exponent laws turn multiplication into addition, division into subtraction, and powers into multiplication.

You can use the rules in either direction. To expand is to rewrite a logarithm of a product, quotient, or power as separate terms. To condense is to combine such terms into one logarithm. Before combining, check that the logarithms have the same base.

\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^p)=p\log_b M

Read $\log_b x$ as the exponent needed on $b$ to produce $x$ .
Product becomes a sum; quotient becomes a difference; a power becomes a coefficient.
The argument of every real logarithm must be positive.

3. Applying the laws carefully

Start by identifying the structure: is the expression a product, a quotient, a power, or a power of a power? Then choose the matching law. Keep the base visible while simplifying. For logarithms, keep the logarithm base visible and check that each argument is positive.

When expanding, a coefficient multiplying a logarithm can be brought in as an exponent on its argument. When condensing, a coefficient becomes an exponent, addition joins arguments by multiplication, and subtraction joins them as a quotient. The order matters: a subtraction in a logarithm expression creates a quotient, not a difference of arguments.

For expressions with variables, restrictions still matter. For example,

\log_5(x-2)

is defined only when

x-2>0

, so

x>2

. A correct use of a law does not remove this restriction. Keep the original expression's allowed values in mind.

A quick check is to test a simple numerical case or convert a logarithm back to exponential form. For instance, if a proposed statement says

\log_3 81=4

, check that

3^4=81

. This helps catch mistakes with bases, signs, and exponents.

c\log_b M=\log_b(M^c)

Match the operation to the correct law before rewriting.
A logarithm difference condenses to a quotient.
State or retain restrictions that make logarithm arguments positive.

4. A reliable process

For a complicated expression, work one change at a time. First simplify powers using the exponent laws. Next expand or condense logarithms only when the arguments and bases allow it. Finally, check whether the rewritten form agrees with the original restrictions.

These laws are especially useful when solving an equation involving powers or logarithms. If you take a logarithm of both sides, use the same valid base on each side and preserve equality. If you combine logarithms, confirm that their bases match. After finding a candidate value, substitute it into the original equation and reject any value that makes a logarithm argument zero or negative.

Do not confuse a power law with a product law. The expression

\log_b(M+N)

does not split into two logarithms. The product law applies to multiplication inside the argument, not addition. Likewise, exponent rules about equal bases cannot be applied to different bases.

Simplify and rewrite in small, checkable steps.
Use logarithm laws only for products, quotients, and powers.
Check a proposed solution in the original expression.

Choosing a law

Structure	Exponent form	Logarithm form
Multiplication	$a^m a^n=a^{m+n}$	$\log_b(MN)=\log_b M+\log_b N$
Division	$\frac{a^m}{a^n}=a^{m-n}$	$\log_b\!\left(\frac{M}{N}\right)=\log_b M-\log_b N$
Power	$(a^m)^n=a^{mn}$	$\log_b(M^p)=p\log_b M$

Worked example

Expand, simplify, and check restrictions

Expand and simplify

\log_2\!\left(\frac{8x^3}{y^2}\right)

, given

x>0

and

y>0

.

Separate the quotient
The argument is a quotient, so the quotient law turns it into a difference of logarithms. The given conditions ensure the variable arguments are positive.
$\log_2\!\left(\frac{8x^3}{y^2}\right)=\log_2(8x^3)-\log_2(y^2)$
Separate the product and powers
The numerator is a product, so split it into a sum. Then move each exponent in the argument to the front as a coefficient.
$\log_2 8+3\log_2 x-2\log_2 y$
Evaluate the numerical logarithm
Because $2^3=8$ , the value of $\log_2 8$ is $3$ . The remaining terms cannot be evaluated without values for $x$ and $y$ .
$3+3\log_2 x-2\log_2 y$

Answer:

3+3\log_2 x-2\log_2 y

Check: For positive

x

and

y

, recombining gives

\log_2(8x^3)-\log_2(y^2)=\log_2\!\left(\frac{8x^3}{y^2}\right)

. Thus the expanded form matches the original.

Common mistakes and how to avoid them

Splitting

\log_b(M+N)

into

\log_b M+\log_b N

.

Correction: The product law applies only to multiplication inside the argument. Addition inside an argument does not split into separate logarithms.

Writing

\log_b(M-N)

as

\log_b M-\log_b N

.

Correction: The subtraction law comes from a quotient:

\log_b(M/N)=\log_b M-\log_b N

. It does not apply to subtraction inside an argument.

Adding exponents when dividing powers with the same base.

Correction: Division subtracts exponents:

a^m/a^n=a^{m-n}

, when the expression is defined.

Ignoring the requirement that a logarithm argument be positive.

Correction: Check the argument before applying a logarithm law or accepting a solution.

Lesson summary

Exponent laws simplify products, quotients, powers, zero exponents, and negative exponents.
A logarithm gives the exponent that produces its argument from its base.
Logarithm laws change products into sums, quotients into differences, and powers into coefficients.
Check bases and positive arguments whenever logarithm laws are used.

Check your understanding

Question 1

Which expression is equivalent to

\log_5(25x)

for

x>0

?

$2+\log_5 x$
$\log_5 25+\log_5 x^2$
$\log_5(25)+x$
$\log_5 25-\log_5 x$

Show answer and explanation

2+\log_5 x

The product law gives

\log_5 25+\log_5 x

. Since

5^2=25

, this is

2+\log_5 x

.

Question 2

Simplify

\frac{3^7}{3^4}

.

$3^3$
$3^{11}$
$3^{7/4}$
$1$

Show answer and explanation

3^3

For division with the same base, subtract the exponents:

3^{7-4}=3^3

.

Key terms

Exponent: A number that tells how many times a base is used as a factor, or indicates a related root or reciprocal.
Base: The number raised to an exponent; in a logarithm, the number whose power is being determined.
Logarithm: The exponent needed on a base to produce a stated positive number.
Argument: The number or expression inside a logarithm.
Expand: Rewrite one logarithm as multiple logarithmic terms using the logarithm laws.
Condense: Combine logarithmic terms into one logarithm using the logarithm laws.

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About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Ontario Grade 12 Mathematics (MHF4U), expectation A1.4. 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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