Understand logarithms and the laws of logarithms

Alberta Grade 12 Mathematics

Relations and Functions

Reading logarithms as exponents and using the basic log laws

A logarithm answers an exponent question. If a quantity grows by repeated multiplication, a logarithm can describe how many equal growth steps are needed to reach a value. You do not need to solve logarithmic equations in this lesson. The goal is to understand what logarithms mean and how their laws let you rewrite expressions accurately.

What you will learn

Explain a logarithm as the exponent that produces a stated number.
Identify the base and argument of a logarithm and state when it is defined.
Use the product, quotient, and power laws to rewrite logarithmic expressions.
Use the laws carefully, keeping the base the same and checking that arguments are positive.

1. Review: powers and exponents

A power is a repeated multiplication. For example,

2^3=8

means that three factors of

2

multiply to make

8

. The base is the number being multiplied, and the exponent tells how many factors are used.

A logarithm reverses this way of asking the question. Instead of asking what number results from a power, it asks what exponent is needed. In

2^3=8

, the exponent question is: “To what power must

2

be raised to get

8

?” The answer is

3

.

The number inside a logarithm is called its argument. The number written below and to the right of the log symbol is its base. The base must be positive and cannot equal

1

. The argument must be positive. These conditions are part of the meaning of a real logarithm.

\log_b a=c\iff b^c=a

A logarithm gives an exponent.
The base is the repeated factor; the argument is the target value.
For a real logarithm, the base is positive and not $1$ , and the argument is positive.

2. Reading and evaluating logarithms

Read

\log_b a

as “the logarithm of

a

to base

b

.” Its value is the exponent that changes

b

into

a

. For example,

\log_5 125=3

because

5^3=125

.

Some familiar values follow directly from powers. A base raised to the exponent

1

stays the same, so

\log_b b=1

. A base raised to the exponent

0

is

1

, so

\log_b 1=0

, as long as the base meets the conditions above.

The argument restriction matters. A logarithm of zero or a negative number is not defined as a real number. For instance,

\log_3(-9)

has no real value. Do not use the product or quotient laws to turn an invalid argument into a valid one.

\log_b b=1,\quad \log_b 1=0

Check a logarithm by converting it to an equivalent power statement.
The argument must be greater than zero.
A logarithm's base must be positive and different from $1$ .

3. The laws of logarithms

The logarithm laws describe how multiplication, division, and powers inside a logarithm can be rewritten. In each law, the logarithms use the same base. The quantities inside the logarithms must be positive.

The product law says that the logarithm of a product is the sum of the logarithms. This fits the exponent meaning: multiplying two powers with the same base adds their exponents. For example,

\log_2(8\cdot4)=\log_2 8+\log_2 4=3+2=5

.

The quotient law says that the logarithm of a quotient is the difference of the logarithms. Dividing powers with the same base subtracts their exponents. The power law says that an exponent on the argument can be brought in front as a multiplier.

These laws can also be used from right to left to combine logarithms, provided the bases match. They do not say that

\log_b(x+y)

can be split into two logarithms. Addition inside an argument is not covered by 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^k)=k\log_b M

Product inside one logarithm becomes addition between logarithms.
Quotient inside one logarithm becomes subtraction between logarithms.
A power on the argument becomes a coefficient in front.
The laws require matching bases and positive arguments.

4. Choosing and checking a rewrite

Before using a law, identify the operation inside the argument. A product calls for the product law; a quotient calls for the quotient law; a power calls for the power law. Keep the base unchanged while applying the law.

A useful way to avoid sign errors is to read the expression aloud. A quotient creates a difference, with the denominator's logarithm subtracted. When expanding a power in a denominator, the quotient law first creates a subtraction; the power law then multiplies the term being subtracted.

You can also combine terms in reverse. A sum of logarithms with a common base can become the logarithm of a product. A difference can become the logarithm of a quotient. This is a rewriting skill, not a method for solving an equation.

Always check the original arguments. If an expression contains

\log_b(x)

, then

x

must be positive wherever the expression is used. In a quotient law, both the numerator and denominator arguments must be positive.

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

Choose the law by looking at the operation inside the argument.
Keep the same base throughout a rewrite.
Check positivity before applying a law.
Do not split a sum inside a logarithm.

Worked example

Find a logarithm from a power

Evaluate

\log_4 64

.

State the exponent question
The logarithm asks which exponent on $4$ produces $64$ .
$4^c=64$
Use a known power
Since $4^3=64$ , the required exponent is $3$ .
$\log_4 64=3$

Answer:

3

Check: Convert the answer back to a power:

4^3=64

, so the value is correct.

Worked example

Expand a logarithm of a quotient and power

Rewrite

\log_3\!\left(\frac{81x^2}{y}\right)

as separate logarithms. Assume

x>0

and

y>0

.

Separate the quotient
The argument is a numerator divided by a denominator, so the quotient law makes a difference of logarithms.
$\log_3(81x^2)-\log_3 y$
Separate the product
The numerator is the product of $81$ and $x^2$ , so the product law separates those factors.
$\log_3 81+\log_3(x^2)-\log_3 y$
Bring down the exponent
The power law changes the exponent $2$ into a coefficient. The assumptions ensure that the logarithms of $x$ and $y$ are defined.
$\log_3 81+2\log_3 x-\log_3 y$
Evaluate the numerical logarithm
Because $3^4=81$ , the first logarithm is $4$ .
$4+2\log_3 x-\log_3 y$

Answer:

4+2\log_3 x-\log_3 y

Check: The quotient gives subtraction, and the square gives a coefficient of

2

. Both variable arguments are positive as required.

Worked example

Condense a sum and difference

Write

2\log_5 m+\log_5 4-\log_5 n

as a single logarithm. Assume

m>0

and

n>0

.

Use the power law in reverse
Move the coefficient $2$ to become an exponent on the argument of the first logarithm.
$\log_5(m^2)+\log_5 4-\log_5 n$
Combine the sum
The first two logarithms have the same base and are added, so their arguments multiply.
$\log_5(4m^2)-\log_5 n$
Combine the difference
A difference of logarithms with the same base becomes the logarithm of the first argument divided by the second.
$\log_5\!\left(\frac{4m^2}{n}\right)$

Answer:

\log_5\!\left(\frac{4m^2}{n}\right)

Check: The combined argument is positive because

m

and

n

are positive, and the denominator is not zero.

Common mistakes and how to avoid them

Treating a sum inside a logarithm like a product and splitting it into two logarithms.

Correction: The product law applies to multiplication inside the argument, not addition. There is no sum law among the laws in this lesson.

Changing the base while applying a product, quotient, or power law.

Correction: Use the laws only when the logarithms have the same base, and retain that base through the rewrite.

Forgetting the minus sign for a denominator when expanding a quotient.

Correction: A quotient becomes the logarithm of the numerator minus the logarithm of the denominator.

Using a logarithm when its argument is zero or negative.

Correction: For real logarithms, every argument must be greater than zero.

Lesson summary

A logarithm is an exponent: $\log_b a=c$ means $b^c=a$ .
The base must be positive and not $1$ , and the argument must be positive.
Products become sums, quotients become differences, and powers become coefficients.
The logarithm laws require matching bases. They do not split addition inside an argument.

Check your understanding

Question 1

What is

\log_2 32

?

$4$
$5$
$16$
$32$

Show answer and explanation

5

Since

2^5=32

, the logarithm is

5

.

Question 2

Which expression is equal to

\log_7(3a)

, for

a>0

?

$\log_7 3+\log_7 a$
$\log_7 3\cdot\log_7 a$
$\log_7 3-\log_7 a$
$3\log_7 a$

Show answer and explanation

\log_7 3+\log_7 a

The argument is a product, so the product law gives the sum of the two logarithms.

Question 3

Which statement is always correct for positive

p

and

q

?

$\log_4(p+q)=\log_4 p+\log_4 q$
$\log_4\!\left(\frac{p}{q}\right)=\log_4 p-\log_4 q$
$\log_4(pq)=\log_4 p-\log_4 q$
$\log_4(p^3)=\log_4 p+3$

Show answer and explanation

\log_4\!\left(\frac{p}{q}\right)=\log_4 p-\log_4 q

The quotient law changes division inside the argument into subtraction between logarithms. The other choices misuse a law.

Key terms

Logarithm: The exponent needed to raise a base to a specified positive value.
Base: The number raised to a power in a logarithm statement.
Argument: The value inside the logarithm symbol.
Product law: The rule that a logarithm of a product can be written as a sum of logarithms with the same base.
Quotient law: The rule that a logarithm of a quotient can be written as a difference of logarithms with the same base.
Power law: The rule that an exponent on a logarithm's argument can be written as a coefficient in front of the logarithm.

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About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Alberta Grade 12 Mathematics (Math 30-2), expectation RF4. 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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RF4 · Understand logarithms and the laws of logarithms

Alberta Grade 12 Mathematics

Reading logarithms as exponents and using the basic log laws

What you will learn

1. Review: powers and exponents

2. Reading and evaluating logarithms

3. The laws of logarithms

4. Choosing and checking a rewrite

Find a logarithm from a power

Expand a logarithm of a quotient and power

Condense a sum and difference

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Key terms

Continue through Math 30-2

About this lesson and its review