Interpret and evaluate logarithms

Ontario Grade 12 Mathematics

Exponential and Logarithmic Functions

Reading a logarithm as a question about an exponent

An exponent tells how many times a base is used as a factor. A logarithm reverses that question: it asks which exponent produces a given number. For example, if a population model uses repeated multiplication by a fixed factor, a logarithm can describe how many multiplication steps are needed to reach a target. In this lesson, you will interpret that question and evaluate logarithms using powers you already know.

What you will learn

Explain what a logarithm asks.
Rewrite between logarithmic and exponential forms.
Evaluate logarithms by identifying the needed exponent.
Check whether a logarithm is defined.

1. Prerequisite bridge: bases and exponents

A power has a base and an exponent. In

3^4

, the base is

3

and the exponent is

4

. The expression means

3 \times 3 \times 3 \times 3

, which equals

81

.

The exponent answers a question about repeated multiplication. If you know the base and exponent, you can find the power. A logarithm reverses this process: if you know the base and the power, it asks for the exponent.

A logarithm needs a positive base that is not

1

. Its input, also called its argument, must be positive. These conditions ensure that the logarithm describes a real exponent in the usual Grade 12 setting.

a^x=y \iff \log_a(y)=x

A power is made from a base and an exponent.
The argument of a logarithm is the number inside the logarithm.
The base must be positive and not equal to $1$ ; the argument must be positive.

2. Plain-language meaning and two forms

Read

\log_a(y)

as “the logarithm of

y

to base

a

.” It asks: “To what exponent must

a

be raised to get

y

?” The answer is the value of the logarithm.

The two forms in the formula above say the same thing. The exponential form shows the base, exponent, and result. The logarithmic form puts the unknown exponent as the value of the logarithm. Moving between the forms is a way to interpret a logarithm, not a different calculation.

For instance, the statement

2^5=32

tells us that the exponent on

2

is

5

when the result is

32

. In logarithmic form, that relationship is written as

\log_2(32)=5

. This small example shows how the notation is read; the guided example later shows a full evaluation.

A logarithm can be negative or zero. A zero value means the base was raised to the power zero. A negative value means a positive base less than or greater than

1

was raised to a negative exponent. The logarithm’s argument still must be positive.

\log_a(y)=x \iff a^x=y

A logarithm’s value is an exponent.
The base in logarithmic form becomes the base in exponential form.
The argument becomes the result, and the logarithm’s value becomes the exponent.

3. Evaluating logarithms and checking their meaning

To evaluate a logarithm exactly, first identify its base and argument. Then ask which exponent on that base gives the argument. Known powers, such as squares, cubes, and powers of ten, are useful. If the exponent is not familiar, rewrite the question in exponential form and use a calculator only when an approximate value is needed.

Common logarithm notation means base

10

:

\log(y)

is another way to write

\log_{10}(y)

. For example, the common logarithm asks which exponent on

10

produces its argument. The notation

\ln(y)

means logarithm to base

e

, where

e

is a positive number approximately equal to

2.718

. At this level, the key idea remains the same: the logarithm gives the exponent.

Before evaluating, check the input. For example,

\log_4(0)

is not defined as a real logarithm because no real exponent on

4

produces zero. Likewise, a negative argument is not allowed in this real-number setting. A base of

1

is also not allowed, since every power of

1

is still

1

and cannot produce other positive arguments.

A useful check is to take your proposed logarithm value and place it as the exponent in the matching power statement. If the power gives the original argument, the evaluation is correct.

\log_{10}(y)=\log(y)

Recognize $\log(y)$ as base $10$ and $\ln(y)$ as base $e$ .
Use a power statement to confirm an exact answer.
Reject logarithms with a nonpositive argument or an invalid base.

4. Guided evaluation and application

Suppose you need to evaluate a logarithm whose base is

5

and whose argument is

\frac{1}{25}

. Start by asking what exponent on

5

produces that fraction. The fraction can be recognized as a reciprocal of a familiar power, so its exponent will be negative.

Writing the relationship in exponential form makes the reasoning visible. Compare the argument with powers of

5

, then select the exponent that produces it. Finally, substitute that exponent back into the power statement to check.

This same interpretation applies in context. If a quantity is multiplied by the same positive factor at each step, a logarithm can represent the number of steps needed to reach a target. The logarithm’s base is the repeated factor, and its argument is the target expressed relative to the starting quantity. Always confirm that the quantities give a positive argument and that the repeated factor is not

1

.

\log_5\left(\frac{1}{25}\right)=-2

Translate the logarithm into a power question before calculating.
A reciprocal of a power corresponds to a negative exponent.
In a repeated-multiplication situation, the logarithm’s value represents the number of steps.

Matching the parts of the two forms

Exponential form	Logarithmic form	Meaning
$a^x=y$	$\log_a(y)=x$	The exponent on $a$ that gives $y$ is $x$ .
$10^x=y$	$\log(y)=x$	The common logarithm gives the exponent on $10$ .
$e^x=y$	$\ln(y)=x$	The natural logarithm gives the exponent on $e$ .

Worked example

Evaluate a logarithm with a fractional argument

Evaluate

\log_5\left(\frac{1}{25}\right)

and explain how the answer can be checked.

Read the question
The logarithm asks which exponent on $5$ produces $\frac{1}{25}$ . Write the question as a power statement.
$5^x=\frac{1}{25}$
Rewrite the argument
Since $25=5^2$ , its reciprocal is $5^{-2}$ . The negative exponent indicates a reciprocal.
$\frac{1}{25}=5^{-2}$
Identify the exponent
The power statement now has the same base and result, so the required exponent is $-2$ .
$x=-2$

Answer:

\log_5\left(\frac{1}{25}\right)=-2

.

Check: Substitute the answer as an exponent:

5^{-2}=\frac{1}{5^2}=\frac{1}{25}

. This reproduces the argument, so the evaluation is correct.

Common mistakes and how to avoid them

Treating the argument as the answer to the logarithm.

Correction: The argument is the number produced by the power. The logarithm’s value is the exponent.

Changing the base or argument when rewriting in exponential form.

Correction: Keep the base as the base, put the logarithm’s value as the exponent, and use the argument as the result.

Assuming a negative logarithm means the argument is negative.

Correction: A negative logarithm value is a negative exponent. The argument must still be positive.

Evaluating a logarithm with argument zero or a negative argument as a real number.

Correction: Check that the argument is positive before evaluating. Such logarithms are not defined in the real-number setting used here.

Lesson summary

A logarithm asks which exponent on a base produces its argument.
The forms $a^x=y$ and $\log_a(y)=x$ express the same relationship.
Evaluate by identifying the exponent, then check by raising the base to that exponent.
The base must be positive and not $1$ , and the argument must be positive.

Check your understanding

Question 1

What is

\log_3(81)

?

$3$
$4$
$27$
correctIndex": 1, "explanation": "The logarithm asks for the exponent on $3$ that gives $81$ . Since $3^4=81$ , the value is $4$ ."

Show answer and explanation

4

The logarithm asks for the exponent on

3

that gives

81

. Since

3^4=81

, the value is

4

.

Question 2

Which logarithm is not defined as a real number?

$\log_2(8)$
$\log_7(1)$
$\log_5(-25)$
correctIndex": 2, "explanation": "A logarithm’s argument must be positive. The argument $-25$ is negative, so the third logarithm is not defined as a real number."

Show answer and explanation

\log_5(-25)

A logarithm’s argument must be positive. The argument

-25

is negative, so the third logarithm is not defined as a real number.

Question 3

Which exponential statement is equivalent to

\log_4(64)=3

?

$4^3=64$
$64^3=4$
$3^4=64$
correctIndex": 0, "explanation": "The base remains $4$ , the logarithm’s value becomes the exponent $3$ , and the argument becomes the result $64$ ."

Show answer and explanation

4^3=64

The base remains

4

, the logarithm’s value becomes the exponent

3

, and the argument becomes the result

64

.

Key terms

Base: The number being raised to an exponent.
Exponent: The number that tells which power of the base is used.
Argument: The number inside a logarithm; it is the result produced by the corresponding power.
Common logarithm: A logarithm with base $10$ , written as $\log(y)$ .
Natural logarithm: A logarithm with base $e$ , written as $\ln(y)$ .

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About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Ontario Grade 12 Mathematics (MHF4U), expectation A1.1. 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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