A logarithm answers the question: 'To what power must the base be raised to get the given number?' If b^c = a, then log_b(a) = c. For example, log_2(8) = 3 because 2^3 = 8.

What is the difference between log and ln?

log (without a base written) usually means log base 10, called the common logarithm. ln means log base e (≈ 2.718), called the natural logarithm. Both follow the same rules, but use different bases.

How are logarithms and exponents related?

Logarithms are the inverse of exponents. The exponential statement b^c = a is equivalent to the logarithmic statement log_b(a) = c. They undo each other: log_b(b^x) = x and b^(log_b(x)) = x.

Why is log(0) undefined?

log_b(0) is undefined because no power of a positive base b can ever equal 0. For example, 10^x is always positive for any real x — it can get very close to 0 but never reach it. Therefore, there is no exponent that gives 0.

What are the main rules of logarithms?

The main rules are: Product Rule: log_b(MN) = log_b(M) + log_b(N). Quotient Rule: log_b(M/N) = log_b(M) − log_b(N). Power Rule: log_b(M^p) = p·log_b(M). Also: log_b(1) = 0, log_b(b) = 1, and the Change of Base Formula: log_b(a) = log(a)/log(b).

How are logarithms used in real life?

Logarithms are used in the pH scale (chemistry), the Richter scale (earthquakes), the decibel scale (sound), population growth models, compound interest calculations, and computer science (algorithm analysis, binary search).

Introduction to Logarithms – Complete Study Guide

Last updated: March 2026 | AP · IB · GCSE · IGCSE · SAT

1. Introduction to Logarithms

What Are Logarithms?

Imagine you already know how to raise numbers to powers. You know that 2³ = 8, that 10² = 100, and that 5⁴ = 625. Exponentiation starts with a base and an exponent, and produces a result. But what if you know the base and the result, and you want to find the exponent? That is exactly what a logarithm does.

A logarithm asks the question: "To what power must I raise the base to get this number?"

For example: "To what power must I raise 2 to get 8?" The answer is 3, because 2³ = 8. We write this as log₂(8) = 3.

In simple terms: logarithms are the inverse of exponents. If exponentiation is the "forward" operation (base × base × base = result), then logarithms are the "reverse" operation (given the result, find how many times the base was multiplied).

Why Are Logarithms Important?

Logarithms are one of the most powerful and widely-used tools in all of mathematics. Here is why they matter:

They tame huge numbers: Logarithms compress enormous numbers into manageable ones. The Richter scale, the pH scale, and the decibel scale all use logarithms to express quantities that range over many orders of magnitude.
They solve exponential equations: If you have an equation like 2^x = 1024, the only way to solve for x algebraically is to take a logarithm: x = log₂(1024) = 10.
They are essential for calculus: The natural logarithm (ln) is one of the most important functions in calculus due to its elegant derivative: d/dx[ln(x)] = 1/x.
They model real-world phenomena: Radioactive decay, population growth, compound interest, and sound intensity all involve exponential relationships, and logarithms are the key to analysing them.
They power computer science: Algorithm complexity is measured using logarithms (e.g., binary search runs in O(log n) time). Information theory uses log₂ to measure bits of information.

Why Students Find Logarithms Difficult at First

If you feel confused when first encountering logarithms, you are in excellent company. Nearly every student struggles initially for these reasons:

The notation is unfamiliar: "log₂(8)" does not look like anything you have seen before. It takes time to read it naturally.
It is an inverse operation: We are used to computing "forward" (given base and exponent, find result). Logarithms work "backwards" (given base and result, find exponent), which requires a mental shift.
Multiple rules to remember: The product rule, quotient rule, power rule, and change of base formula can feel overwhelming at first.

The good news is that logarithms are not actually difficult. They just require practice and the right way of thinking. By the end of this guide, you will find them as natural as multiplication.

💡 Teacher Tip: Every time you see log_b(a), mentally ask: "b to what power gives a?" Train this reflex until it is automatic. This single habit will make every logarithm problem easier.

2. Understanding the Meaning of a Logarithm

The Definition

log_b(a) = c means b^c = a
b = base | a = argument | c = exponent (the answer)

Let us break this apart piece by piece:

The base (b): The number being raised to a power. It must be positive and not equal to 1 — i.e., b > 0 and b ≠ 1.
The argument (a): The number you are taking the logarithm of. It must be positive — i.e., a > 0. You cannot take the log of zero or a negative number in real math.
The answer (c): The exponent — the power to which the base must be raised. This can be any real number (positive, negative, or zero).

Think of it this way: The logarithm is asking, "The base raised to what gives me the argument?" The answer is the exponent.

Simple Examples — Step by Step

Example 1 — log₂(8) = ?

Question: 2 to what power gives 8?

Think: 2¹ = 2, 2² = 4, 2³ = 8 ✓

Answer: log₂(8) = 3

Example 2 — log₁₀(100) = ?

Question: 10 to what power gives 100?

Think: 10¹ = 10, 10² = 100 ✓

Answer: log₁₀(100) = 2

Example 3 — log₃(27) = ?

Question: 3 to what power gives 27?

Think: 3¹ = 3, 3² = 9, 3³ = 27 ✓

Answer: log₃(27) = 3

Example 4 — log₅(1) = ?

Question: 5 to what power gives 1?

Think: Any number to the power 0 equals 1. So 5⁰ = 1 ✓

Answer: log₅(1) = 0

This is true for every base: log_b(1) = 0 always.

Example 5 — log₄(1/16) = ?

Question: 4 to what power gives 1/16?

Think: 1/16 = 1/4² = 4⁻²

Answer: log₄(1/16) = −2

Logarithms can be negative! A negative log value just means the argument is a fraction (less than 1).

3. Connection Between Exponents and Logarithms

Understanding that logarithms and exponents are inverse operations is the single most important concept in this topic. They undo each other, just like addition undoes subtraction and multiplication undoes division.

Exponential Form ↔ Logarithmic Form

b^c = a ↔ log_b(a) = c

These two statements say exactly the same thing in different notation. Learning to convert fluently between them is the key skill.

Conversion Examples

Convert to Logarithmic Form

a) 2³ = 8 → log₂(8) = 3

b) 10² = 100 → log₁₀(100) = 2

c) 5⁰ = 1 → log₅(1) = 0

d) 3⁻² = 1/9 → log₃(1/9) = −2

e) e¹ = e → ln(e) = 1

Convert to Exponential Form

a) log₂(32) = 5 → 2⁵ = 32

b) log₁₀(1000) = 3 → 10³ = 1000

c) log₇(1) = 0 → 7⁰ = 1

d) ln(1) = 0 → e⁰ = 1

The Inverse Relationship — Cancelling Each Other

Because logarithms and exponents are inverses, they cancel each other out when applied in sequence:

log_b(b^x) = x and b^log_b(x) = x

These two identities are extremely powerful for simplification:

log₂(2⁵) = 5 (the log and the base-2 exponent cancel)
10^{log₁₀(50)} = 50 (the exponent and the log cancel)
e^ln(7) = 7 (e and ln are inverses)
ln(e⁻³) = −3 (ln undoes the e)

4. Common Types of Logarithms

While you can have a logarithm with any valid base, two bases are used so frequently that they have their own special notation.

4.1 Common Logarithm (Base 10)

log(x) = log₁₀(x)

When you see "log" written without a base, it almost always means base 10 (this is the convention in most countries, textbooks, and calculators). It is called the common logarithm because base 10 aligns with our decimal number system.

log(10) = 1 (because 10¹ = 10)
log(100) = 2 (because 10² = 100)
log(1000) = 3 (because 10³ = 1000)
log(1) = 0 (because 10⁰ = 1)
log(0.01) = −2 (because 10⁻² = 0.01)

💡 Calculator Note: The "log" button on your scientific calculator computes log base 10. When a question says "evaluate log(500)," press the log button, type 500, and you get approximately 2.699.

4.2 Natural Logarithm (Base e)

ln(x) = log_e(x) where e ≈ 2.71828...

The number e (Euler's number) is one of the most important constants in mathematics, approximately equal to 2.71828. The natural logarithm, written as ln, is the logarithm with base e. It is called "natural" because it arises naturally in calculus, continuous growth, and many areas of physics.

ln(e) = 1 (because e¹ = e)
ln(e²) = 2 (because e² ≈ 7.389)
ln(1) = 0 (because e⁰ = 1)
ln(e⁻¹) = −1

💡 Calculator Note: The "ln" button on your calculator computes the natural logarithm. When a question says "evaluate ln(5)," press ln, type 5, and you get approximately 1.609.

4.3 Logarithms with Other Bases

You can have a logarithm with any positive base (other than 1). Some examples students often encounter:

log₂(x) — binary logarithm, crucial in computer science.
log₃(x), log₅(x), log₇(x) — used in various exam questions.

Most calculators only have log (base 10) and ln (base e) buttons. To evaluate other bases, you use the Change of Base Formula (covered in Section 6).

5. Evaluating Basic Logarithms

The systematic method for evaluating any logarithm is: (1) identify the base and argument, (2) rewrite as an exponential equation, (3) solve for the exponent. Let us practise this process.

Example 1 — Evaluate log₂(16)

Step 1 – Identify: Base = 2, Argument = 16.

Step 2 – Rewrite: 2^x = 16

Step 3 – Solve: 2⁴ = 16, so x = 4.

Answer: log₂(16) = 4

Example 2 — Evaluate log₁₀(1000)

Step 1 – Identify: Base = 10, Argument = 1000.

Step 2 – Rewrite: 10^x = 1000

Step 3 – Solve: 10³ = 1000, so x = 3.

Answer: log₁₀(1000) = 3

Example 3 — Evaluate log₄(1/16)

Step 1 – Identify: Base = 4, Argument = 1/16.

Step 2 – Rewrite: 4^x = 1/16

Step 3 – Solve: 1/16 = 1/4² = 4⁻², so x = −2.

Answer: log₄(1/16) = −2

Example 4 — Evaluate ln(e³)

Step 1 – Identify: Base = e, Argument = e³.

Step 2 – Use inverse property: log_b(b^x) = x, so ln(e³) = 3.

Answer: ln(e³) = 3

Example 5 — Evaluate log₅(125)

Step 1 – Identify: Base = 5, Argument = 125.

Step 2 – Rewrite: 5^x = 125

Step 3 – Solve: 5³ = 125, so x = 3.

Answer: log₅(125) = 3

Example 6 — Evaluate log₈(2)

Step 1 – Identify: Base = 8, Argument = 2.

Step 2 – Rewrite: 8^x = 2

Step 3 – Solve: 8 = 2³, so 8^x = (2³)^x = 2^3x = 2¹. Therefore 3x = 1, so x = 1/3.

Answer: log₈(2) = 1/3

Logarithms can be fractions! This means 8^1/3 = ∛8 = 2.

6. Properties and Rules of Logarithms

Just as exponents have laws (product rule, quotient rule, power rule), logarithms have corresponding rules. These are derived directly from the exponent laws, using the fact that logarithms are inverse functions of exponents.

6.1 Product Rule

log_b(M × N) = log_b(M) + log_b(N)

In words: The log of a product equals the sum of the logs. Multiplication inside the log becomes addition outside.

Why it works: If b^p = M and b^q = N, then M × N = b^p × b^q = b^p+q. Taking log base b: log_b(MN) = p + q = log_b(M) + log_b(N).

Example — Product Rule

Expand: log₂(8 × 4)

= log₂(8) + log₂(4) = 3 + 2 = 5

Check: 8 × 4 = 32, and log₂(32) = 5. ✓

6.2 Quotient Rule

log_b(M / N) = log_b(M) − log_b(N)

In words: The log of a quotient equals the difference of the logs. Division inside the log becomes subtraction outside.

Example — Quotient Rule

Expand: log₁₀(1000 / 10)

= log₁₀(1000) − log₁₀(10) = 3 − 1 = 2

Check: 1000/10 = 100, and log₁₀(100) = 2. ✓

6.3 Power Rule

log_b(M^p) = p · log_b(M)

In words: The log of a power equals the exponent times the log. The exponent "comes down" in front of the log as a multiplier.

Example — Power Rule

Expand: log₃(9⁵)

= 5 · log₃(9) = 5 × 2 = 10

Check: 9⁵ = 59,049, and 3¹⁰ = 59,049. ✓

6.4 Log of 1

log_b(1) = 0 (for any valid base b)

Why: b⁰ = 1 for any non-zero b. Since the log asks "b to what power gives 1?", the answer is always 0.

6.5 Log of the Base

log_b(b) = 1 (for any valid base b)

Why: b¹ = b. The log asks "b to what power gives b?", and the answer is always 1.

6.6 Change of Base Formula

log_b(a) = log(a) / log(b) = ln(a) / ln(b)

Why this matters: Your calculator only has buttons for log (base 10) and ln (base e). If you need to evaluate log₃(50), use: log₃(50) = log(50) / log(3) ≈ 1.699 / 0.477 ≈ 3.561.

Example — Change of Base

Evaluate: log₇(200)

= log(200) / log(7) = 2.3010 / 0.8451 ≈ 2.723

7. Domain and Restrictions of Logarithms

Logarithms have strict rules about what values are allowed. Understanding these restrictions is crucial for solving equations and graphing.

Rule 1: The Argument Must Be Positive

log_b(a) is defined only when a > 0

Why: There is no real number x such that b^x is zero or negative (when b is positive). Since exponential functions with positive bases always produce positive outputs, the "input" to a logarithm (the argument) must also be positive.

❌ Invalid Logarithmic Expressions

log(0) — undefined. No power of 10 equals 0.

log(−5) — undefined. No power of 10 equals −5.

ln(−1) — undefined in real numbers.

Rule 2: The Base Must Be Positive and Not 1

Base b must satisfy: b > 0 and b ≠ 1

Why b > 0: Negative bases create problems with non-integer exponents (e.g., (−2)^1/2 is not real). So we require positive bases.

Why b ≠ 1: 1 raised to any power is always 1. So log₁(a) would only work if a = 1, and even then, every exponent would be valid. The function is not well-defined, so base 1 is excluded.

Practice — Valid or Invalid?

a) log₃(9) — ✅ Valid (base 3 > 0 and ≠ 1, argument 9 > 0)

b) log₋₂(8) — ❌ Invalid (negative base)

c) log₅(0) — ❌ Invalid (argument is 0)

d) ln(−3) — ❌ Invalid (negative argument)

e) log₁(10) — ❌ Invalid (base is 1)

f) log_0.5(4) — ✅ Valid (base 0.5 > 0 and ≠ 1, argument 4 > 0)

8. Solving Simple Logarithmic Equations

The key technique for solving logarithmic equations is to rewrite the equation in exponential form. Once it is in exponential form, it becomes a standard algebra problem.

Example 1 — Solve log₂(x) = 4

Step 1 – Rewrite in exponential form: 2⁴ = x

Step 2 – Evaluate: x = 16

Step 3 – Check: log₂(16) = 4 ✓ (because 2⁴ = 16)

Answer: x = 16

Example 2 — Solve log₁₀(x) = 3

Step 1 – Rewrite: 10³ = x

Step 2 – Evaluate: x = 1000

Answer: x = 1000

Example 3 — Solve ln(x) = 2

Step 1 – Rewrite: e² = x

Step 2 – Evaluate: x = e² ≈ 7.389

Answer: x = e² ≈ 7.389

Example 4 — Solve log₃(x + 1) = 2

Step 1 – Rewrite: 3² = x + 1

Step 2 – Simplify: 9 = x + 1, so x = 8

Step 3 – Check domain: x + 1 = 9 > 0 ✓

Answer: x = 8

Example 5 — Solve log₅(2x − 3) = 1

Step 1 – Rewrite: 5¹ = 2x − 3

Step 2 – Simplify: 5 = 2x − 3 → 2x = 8 → x = 4

Step 3 – Check domain: 2(4) − 3 = 5 > 0 ✓

Answer: x = 4

⚠️ Always Check Domain: After solving a logarithmic equation, substitute your answer back to make sure the argument of the logarithm is positive. If it is not, the solution is extraneous (invalid) and must be rejected.

9. Graph of a Logarithmic Function

Understanding the graph of y = log_b(x) is essential for exams and for building intuition about how logarithms behave.

Key Features of y = log_b(x) (when b > 1)

Domain: x > 0 (only positive x-values are allowed).
Range: All real numbers (−∞, +∞). The output can be any real number.
Vertical asymptote: x = 0 (the y-axis). The curve approaches the y-axis but never touches or crosses it.
x-intercept: (1, 0). Because log_b(1) = 0 for any base b.
The point (b, 1): Because log_b(b) = 1. For log base 10: the point (10, 1). For ln: the point (e, 1) ≈ (2.718, 1).
Shape: The curve increases slowly to the right (it grows, but very slowly). It is concave down — it curves downward as it goes right.
Passes through: (1, 0), (b, 1), (b², 2), (b³, 3), etc.

The Inverse Relationship with Exponential Graphs

The graph of y = log_b(x) is the reflection of y = b^x in the line y = x. This makes perfect sense because logarithms and exponents are inverses — if you reflect any function across y = x, you get its inverse.

Feature	y = b^x	y = log_b(x)
Domain	All real numbers	x > 0
Range	y > 0	All real numbers
Asymptote	y = 0 (horizontal)	x = 0 (vertical)
Key point	(0, 1)	(1, 0)
Passes through	(1, b)	(b, 1)

💡 Graph Tip for Exams: To sketch y = log_b(x), plot three key points: (1, 0), (b, 1), and (1/b, −1). Draw a vertical asymptote at x = 0. Then draw a smooth curve through these points that increases slowly to the right and drops steeply near the asymptote on the left.

What About 0 < b < 1?

When the base is between 0 and 1 (like 1/2), the logarithmic function is decreasing. The graph is reflected horizontally compared to bases greater than 1. For example, log_1/2(x) decreases as x increases. However, it still has the same domain (x > 0), range (all real numbers), and vertical asymptote at x = 0.

10. Real-Life Applications of Logarithms

Logarithms are not abstract mathematical oddities — they are used constantly in science, technology, and everyday life. Here are some of the most important applications.

10.1 The pH Scale (Chemistry)

The pH of a solution is defined as pH = −log₁₀[H⁺], where [H⁺] is the hydrogen ion concentration. This converts very small concentrations (like 0.0000001) into manageable numbers (like 7). The pH scale runs from 0 to 14, where 7 is neutral (pure water), values below 7 are acidic, and values above 7 are alkaline. Every single pH calculation in chemistry is a direct application of common logarithms.

10.2 The Richter Scale (Earthquakes)

Earthquake magnitude is measured on the Richter scale, which is logarithmic. Each whole-number increase represents a tenfold increase in ground shaking and approximately 31.6 times more energy released. A magnitude 6 earthquake is 10 times more powerful than a magnitude 5, and 100 times more powerful than a magnitude 4. Without logarithms, comparing earthquake strengths would require cumbersome numbers spanning many orders of magnitude.

10.3 Sound Intensity (Decibels)

Sound level in decibels (dB) is calculated using L = 10 log₁₀(I/I₀), where I is the sound intensity and I₀ is the threshold of human hearing. Normal conversation is about 60 dB, a rock concert is about 110 dB, and the threshold of pain is about 130 dB. Because the decibel scale is logarithmic, an increase of 10 dB represents a 10-fold increase in sound intensity. Our ears perceive this as roughly "twice as loud."

10.4 Population Growth and Compound Interest

Exponential growth models use the formula N = N₀ e^rt. To find when a population reaches a certain size, you solve for t by taking the natural logarithm: t = ln(N/N₀) / r. Similarly, the time required for an investment to double at a given interest rate is found using logarithms: t = ln(2) / r ≈ 0.693/r. These financial and biological calculations are impossible without logarithms.

10.5 Computer Science

Logarithms are fundamental to computer science. The binary search algorithm finds an item in a sorted list of n elements in log₂(n) steps. Sorting algorithms like merge sort run in O(n log n) time. Data compression, encryption, and information theory all heavily use logarithms. The concept of a "bit" itself is defined using log₂ — the number of bits needed to represent n possibilities is log₂(n).

10.6 Astronomy — Stellar Brightness

The apparent magnitude of stars is measured on a logarithmic scale. A difference of 5 magnitudes corresponds to a factor of 100 in brightness. This means each magnitude difference is a factor of 100^1/5 ≈ 2.512. The brightest star visible from Earth (Sirius) has a magnitude of about −1.46, while the faintest stars visible to the naked eye have a magnitude of about +6. Without logarithms, the vast range of stellar brightnesses would be impossible to express in a convenient numerical system.

🧪 Practical Example — Calculating pH

Problem: A solution has a hydrogen ion concentration of [H⁺] = 3.2 × 10⁻⁵ mol/L. Find the pH.

Formula: pH = −log₁₀(3.2 × 10⁻⁵)

Step 1: log(3.2 × 10⁻⁵) = log(3.2) + log(10⁻⁵) = 0.505 + (−5) = −4.495

Step 2: pH = −(−4.495) = 4.495

This is acidic (below 7), similar to tomato juice.

💰 Practical Example — Doubling Time for an Investment

Problem: How long does it take an investment to double at 5% annual interest, compounded continuously?

Formula: t = ln(2) / r

Solution: t = ln(2) / 0.05 = 0.6931 / 0.05 = 13.86 years

This is also known as the "Rule of 70" — divide 70 by the interest rate percentage to get the approximate doubling time.

11. Common Student Mistakes

After twenty years of teaching logarithms, these are the errors I see most frequently. Study them carefully — avoiding these mistakes is worth just as many marks as knowing the rules.

❌ Mistake 1: Thinking log(a + b) = log(a) + log(b)

Wrong: log(3 + 5) = log(3) + log(5)

Why it is wrong: The product rule says log(a × b) = log(a) + log(b). The rule involves multiplication, not addition. There is no logarithm rule for sums.

Correct: log(3 + 5) = log(8) ≈ 0.903, while log(3) + log(5) = log(15) ≈ 1.176.

❌ Mistake 2: Confusing the Base and the Argument

Wrong: Reading log₂(8) as "8 to the power of 2"

Correct: log₂(8) means "2 to what power gives 8?"

❌ Mistake 3: Forgetting the Argument Must Be Positive

Wrong: Solving log₂(x − 5) = 3 without checking the domain.

Correct: Always ensure the argument of the log is positive.

❌ Mistake 4: log(a/b) ≠ log(a) / log(b)

Wrong: log(100/10) = log(100) / log(10)

Correct: log(100/10) = log(100) − log(10)

❌ Mistake 5: Misapplying the Power Rule

Wrong: log(x)² = 2log(x)

Correct: log(x²) = 2log(x)

❌ Mistake 6: Using log Without Knowing the Base

Correct: log = base 10, ln = base e.

12. Exam Tips and Strategies

🎯 Strategy 1

Always read log_b(a) as “b to what power equals a?”.

🎯 Strategy 2

Convert to exponential form when solving equations.

🎯 Strategy 3

Multiplication → Product rule
Division → Quotient rule
Exponent → Power rule

🎯 Strategy 4

Always check domain restrictions.

🎯 Strategy 5

Remember log_b(1)=0 and log_b(b)=1.

🧮 Interactive Logarithm Calculator

Logarithm Calculator

Enter a base and argument to compute log_b(a). See the logarithmic value and the equivalent exponential statement.

Base (b):

Argument (a):

14. Summary of Key Points

Concept / Rule	Formula	Key Idea
Logarithm Definition	log_b(a) = c ⟺ b^c = a	"b to what power gives a?"
Product Rule	log_b(MN) = log_b(M) + log_b(N)	Multiplication → Addition
Quotient Rule	log_b(M/N) = log_b(M) − log_b(N)	Division → Subtraction
Power Rule	log_b(M^p) = p · log_b(M)	Exponent comes down as multiplier
Log of 1	log_b(1) = 0	Any base to the power 0 = 1
Log of Base	log_b(b) = 1	Any base to the power 1 = itself
Change of Base	log_b(a) = log(a)/log(b)	Convert to base 10 or e for calculator
Inverse Properties	log_b(b^x) = x & b^log_b(x) = x	Log and exponent cancel each other
Domain Restriction	Argument must be > 0	log(0) and log(negative) are undefined
Common Log	log(x) = log₁₀(x)	Base 10, used in science and engineering
Natural Log	ln(x) = log_e(x)	Base e ≈ 2.718, used in calculus

📌 Final Reminders

A logarithm asks: "base to what power gives the argument?"
Logarithms and exponents are inverses — they cancel each other.
"log" = base 10, "ln" = base e. Know the difference.
The argument must always be positive — check domain on every equation.
log(a × b) = log(a) + log(b), but log(a + b) ≠ log(a) + log(b). No sum rule!
When stuck, convert to exponential form — it makes most problems straightforward.
Use the change of base formula to evaluate unusual bases on your calculator.
Always show working and check answers for full exam marks.

🎓 You have completed the Introduction to Logarithms Study Guide!
Bookmark this page for quick revision before your exams. Good luck! 🍀