Teacher Guide: Properties of Logarithms

Learn the basic properties of logarithms: the log of 1, the log of the base, the two inverse properties and the one-to-one property.

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10 questions on Logarithms. Students join with a name, you see everyone's score.

For Teachers

Learning Objectives

Evaluate $\log_b 1$ and $\log_b b$ for any allowed base
Use the inverse properties $\log_b(b^x) = x$ and $b^{\log_b x} = x$ to simplify expressions
Solve equations of the form $\log_b M = \log_b N$ with the one-to-one property
Recognize undefined logarithms and reject candidates that make an argument zero or negative
Name the product, quotient and power rules and connect them to the basic properties

Prerequisites

• Definition of a Logarithm: converting between $\log_b x = y$ and $b^y = x$
• Exponent rules, including $b^0 = 1$ and negative exponents
• Solving linear equations and simple quadratic equations by factoring

Discussion Starters

1. Every graph $y = \log_b x$ passes through the point $(1, 0)$ . Which property explains this?
2. Why must the bases match before $b^{\log_b x}$ simplifies to $x$ ? Try $2^{\log_4 16}$ .
3. Two different positive numbers never have the same logarithm in the same base. How does the graph show this?
4. Can a logarithm have a negative value? Can it have a negative argument? What is the difference?

Common Misconceptions

$\log_b 1 = 1$ because 'the log of one is one'

Remediation: Go back to the question the logarithm asks: which power of $b$ gives $1$ ? Let students test $2^1$ , $2^0$ and see that only the exponent $0$ works.

A logarithm and a power always cancel, whatever the bases

Remediation: Compare $3^{\log_3 9} = 9$ with $2^{\log_3 9} = 2^2 = 4$ . Have students circle both bases before they cancel anything.

Every solution of $M = N$ is a solution of $\log_b M = \log_b N$

Remediation: Solve $\log_4(x^2 - 4) = \log_4(3x)$ together and test $x = -1$ in the original equation. Make the domain check a fixed last step.

Differentiation Ideas

For Struggling Students:

• Start with base $10$ and $2$ only and keep a table of powers ( $2^1$ to $2^{10}$ ) on the desk
• Practice each property separately before mixing them
• Read every logarithm out loud as a question: which power of $b$ gives $x$ ?

For On-Level Students:

• Mix all five properties in one set of evaluation problems
• Solve one-to-one equations with linear arguments and check the arguments
• Use the graph to explain why $\log_b 1 = 0$ and $\log_b b = 1$ hold for every base

For Advanced Students:

• Solve one-to-one equations that lead to quadratics and reject the candidates outside the domain
• Prove $b^{\log_b x} = x$ and $\log_b(b^x) = x$ from the definition of the logarithm
• Preview how the product rule follows from $b^m \cdot b^n = b^{m+n}$

Standards Alignment

HSF-BF.B.5 (CCSS.MATH.CONTENT.HSF.BF.B.5)
Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents
HSF-LE.A.4 (CCSS.MATH.CONTENT.HSF.LE.A.4)
Express the solution to an exponential model as a logarithm and evaluate the logarithm

Lesson Resources

visualLogarithm Graph With a Base Slider
Students change the base $b$ and see that every curve $y = \log_b x$ passes through $(1, 0)$ and $(b, 1)$
activityProperty Sort
Cards with expressions like $\log_6 1$ , $5^{\log_5 8}$ and $\log_2(2^9)$ are sorted by the property they use and then evaluated
worksheetDefined or Undefined?
Students decide which logarithms exist and check candidate solutions of one-to-one equations

Lesson Content

Everything students see: definition, examples, common mistakes, applications. Tap to open.

Definition

The basic properties of logarithms all come from one fact: a logarithm is an exponent. For a base

b > 0

with

b \neq 1

\log_b x = y \iff b^y = x

The five basic properties

1.Log of 1: $\log_b 1 = 0$ , because $b^0 = 1$
2.Log of the base: $\log_b b = 1$ , because $b^1 = b$
3.Inverse property (log of a power of the base): $\log_b(b^x) = x$ for every real number $x$
4.Inverse property (base raised to a log): $b^{\log_b x} = x$ for every $x > 0$
5.One-to-one property: if $\log_b M = \log_b N$ (with $M > 0$ , $N > 0$ ), then $M = N$

What is not allowed: the argument of a logarithm must be positive.

\log_b 0

and

\log_b(-5)

are undefined, because

b^y

is positive for every exponent

y

Properties 3 and 4 say that

y = \log_b x

and

y = b^x

are inverse functions: one undoes the other.

Overview: the three rules for combining logarithms (each one has its own lesson)

Rule	Formula	Lesson
Product rule	$\log_b(MN) = \log_b M + \log_b N$	Product Rule for Logarithms
Quotient rule	$\log_b \frac{M}{N} = \log_b M - \log_b N$	Quotient Rule for Logarithms
Power rule	$\log_b(M^p) = p \log_b M$	Power Rule for Logarithms

In this lesson we practice the five basic properties. The three rules in the table are taught step by step in their own lessons.

Worked Examples

Evaluate $\log_9 1 + \log_9 9$ .

Evaluate $\log_9 1$

Which power of $9$ gives $1$ ? $9^0 = 1$ → $\log_9 1 = 0$

Evaluate $\log_9 9$

Which power of $9$ gives $9$ ? $9^1 = 9$ → $\log_9 9 = 1$

Add the results

$0 + 1$

Answer: $\log_9 1 + \log_9 9 = 1$

Common Mistakes

Writing $\log_b 1 = 1$ or $\log_b 1 = b$

Why it's wrong: The logarithm is the exponent, not the result. The question is: which power of $b$ gives $1$ ?

Correct: $b^0 = 1$ , so $\log_b 1 = 0$ for every allowed base. For example $\log_7 1 = 0$ .

Thinking $\log_b 0 = 0$ or that the log of a negative number is a negative number

Why it's wrong: A positive base raised to any power is always positive, so no exponent gives $0$ or a negative number.

Correct: $\log_b 0$ and $\log_b(-4)$ are undefined. The argument must be greater than $0$ , while the value of a logarithm can be negative: $\log_2 \frac{1}{8} = -3$ .

Cancelling a power and a logarithm with different bases: $2^{\log_3 9} = 9$

Why it's wrong: $b^{\log_b x} = x$ only works when the base of the power and the base of the logarithm are the same number.

Correct: Here the bases differ: $\log_3 9 = 2$ , so $2^{\log_3 9} = 2^2 = 4$ , not $9$ .

Using the one-to-one property and not checking the arguments

Why it's wrong: Setting the arguments equal can produce a value that makes an argument zero or negative, where the original logarithm does not exist.

Correct: Always put each candidate back into every argument. Keep it only if all arguments are positive.

Why It Matters

The basic properties are the tools you use every time you work with logarithms:

Evaluating without a calculator: $\log_2 64 = \log_2(2^6) = 6$ in one line
Solving equations: the inverse properties remove a logarithm or an exponential, so $\log_3 x = 4$ becomes $x = 3^4 = 81$
Comparing: the one-to-one property turns $\log_5(2x + 1) = \log_5 11$ into the simple equation $2x + 1 = 11$
Science scales: decibels, pH and the Richter scale all use $\log$ . A sound at the hearing threshold has $10 \log 1 = 0$ decibels
Avoiding errors: knowing that $\log_b 0$ and logs of negative numbers are undefined tells you which solutions to throw away

The product, quotient and power rules (and the change of base formula) are all built on these five properties.

Real World Applications

Sound Level in Decibels

The loudness of a sound is $L = 10 \log \frac{I}{I_0}$ decibels, where $I_0$ is the quietest sound a person can hear. The basic properties make the calculation quick.

Example:

At the hearing threshold $I = I_0$ , so $L = 10 \log 1 = 10 \cdot 0 = 0$ decibels.

1Try It Yourself

A rock concert has an intensity of $I = 10^{11} \cdot I_0$ , so $\frac{I}{I_0} = 10^{11}$ .

What is the sound level of the concert in decibels?

Step 1: Write the mathematical expression

Compute $L = 10 \log(10^{11})$ :

Acidity and pH

Chemists measure acidity with $\text{pH} = -\log[H^+]$, where $[H^+]$ is the hydrogen ion concentration in moles per liter. The inverse property turns a pH back into a concentration.

Example:

Pure water has $[H^+] = 10^{-7}$ , so $\text{pH} = -\log(10^{-7}) = -(-7) = 7$ .

2Try It Yourself

Lemon juice has a pH of $2$ , so $-\log[H^+] = 2$ and $\log[H^+] = -2$ .

What is the hydrogen ion concentration $[H^+]$ of lemon juice?

Step 1: Write the mathematical expression

Use $10^{\log x} = x$ and compute $[H^+] = 10^{-2}$ :

Rounds in a Knockout Tournament

In a knockout tournament half of the teams drop out in every round. With $2^n$ teams there are $\log_2(2^n) = n$ rounds until one winner is left.

Example:

With $16 = 2^4$ teams there are $\log_2 16 = \log_2(2^4) = 4$ rounds.

3Try It Yourself

A school chess tournament starts with $128$ players. Every game has a winner, and the loser drops out.

How many rounds are needed to find the champion?

Step 1: Write the mathematical expression

Compute $\log_2 128$ :

Key Takeaways

1 $\log_b 1 = 0$ because $b^0 = 1$ , and $\log_b b = 1$ because $b^1 = b$
2Inverse property: $\log_b(b^x) = x$ for every real $x$
3Inverse property: $b^{\log_b x} = x$ for every $x > 0$ , but only when both bases are the same
4One-to-one property: if $\log_b M = \log_b N$ , then $M = N$ ; always check that the arguments are positive
5 $\log_b 0$ and the logarithm of a negative number are undefined, while the value of a logarithm may be negative
6The product, quotient and power rules build on these properties and are taught in their own lessons

Frequently Asked Questions

Why is $\log_b 1$ always $0$ ?

Any allowed base raised to the power

0

equals

1

2^0 = 1

10^0 = 1

e^0 = 1

. The logarithm asks for that exponent, so the answer is always

0

Can the value of a logarithm be negative?

Yes. Only the argument has to be positive. For example

\log_{10} 0.01 = -2

because

10^{-2} = 0.01

. The argument

0.01

is positive, the value

-2

is negative.

Why does $b^{\log_b x} = x$ need $x > 0$ , while $\log_b(b^x) = x$ works for every $x$ ?

\log_b(b^x)

the argument

b^x

is always positive, so every real

x

is allowed. In

b^{\log_b x}

the logarithm

\log_b x

must exist first, and that needs

x > 0

Where are the product, quotient and power rules?

Each has its own lesson: Product Rule for Logarithms, Quotient Rule for Logarithms and Power Rule for Logarithms. They use the properties from this lesson, so it helps to master these first.

Glossary

Logarithm: The exponent $y$ with $b^y = x$ , written $\log_b x = y$
Base: The number $b$ in $\log_b x$ ; it must satisfy $b > 0$ and $b \neq 1$
Argument: The number $x$ in $\log_b x$ ; it must be positive
Inverse properties: $\log_b(b^x) = x$ and $b^{\log_b x} = x$ : the logarithm and the exponential function with the same base undo each other
One-to-one property: If $\log_b M = \log_b N$ , then $M = N$ : different positive arguments never give the same logarithm
Inverse function: A function that reverses another one; $y = \log_b x$ is the inverse function of $y = b^x$

Formula Card

Log of 1

\log_b 1 = 0

Because $b^0 = 1$

Log of the base

\log_b b = 1

Because $b^1 = b$

Inverse property 1

\log_b(b^x) = x

For every real number $x$

Inverse property 2

b^{\log_b x} = x

For every $x > 0$, same base on both levels

One-to-one property

\log_b M = \log_b N \Rightarrow M = N

Check that $M > 0$ and $N > 0$

Domain

\log_b x

exists only for

x > 0

$\log_b 0$ and logs of negative numbers are undefined

Teacher Guide: Properties of Logarithms

Use this lesson with your class

For Teachers

For Struggling Students:

For On-Level Students:

For Advanced Students:

Lesson Content

Definition

Worked Examples

Common Mistakes

Why It Matters

Real World Applications

Sound Level in Decibels

Acidity and pH

Rounds in a Knockout Tournament

Key Takeaways

Frequently Asked Questions

Glossary

Formula Card

Log of 1

Log of the base

Inverse property 1

Inverse property 2

One-to-one property

Domain

More in This Topic

Introduction to Logarithms

Definition of a Logarithm

Common Logarithms (Base 10)

Natural Logarithms (ln)