Logarithms

What Is a Logarithm?

A logarithm answers the question: to what power must a base be raised to produce a given number?

The notation log₍b₎(x) = y means bʸ = x.

• b is the base
• x is the argument (what you're taking the log of)
• y is the result (the exponent)

Example: log₂(32) = 5, because 2⁵ = 32.

Logarithms and exponentiation are inverse operations — each undoes the other:

• If 10² = 100, then log₁₀(100) = 2
• If 3³ = 27, then log₃(27) = 3

Common Bases

Base 10 — common logarithm

Written as log(x) without a base shown, it means log₁₀(x). This appears on most calculators as the log button.

• log(10) = 1
• log(100) = 2
• log(1000) = 3
• log(0.1) = −1

Base e — natural logarithm

Written as ln(x), where e ≈ 2.718 (Euler's number). This is fundamental in calculus, science, and finance. It appears as the ln button on a calculator.

• ln(e) = 1
• ln(e²) = 2
• ln(1) = 0

Base 2 — binary logarithm

Written as log₂(x). Essential in computing — it tells you how many bits are needed to represent a number.

• log₂(2) = 1
• log₂(4) = 2
• log₂(1024) = 10

Two Key Values to Memorise

For any valid base b (where b is positive and not equal to 1):

• log₍b₎(1) = 0 — because b⁰ = 1 for any base
• log₍b₎(b) = 1 — because b¹ = b

These are useful checkpoints when evaluating logs.

Laws of Logarithms

The three laws let you simplify expressions with logs, just like the laws of indices simplify powers.

Product Law

log(ab) = log(a) + log(b)

Logs turn multiplication into addition.

Example: log(3) + log(4) = log(3 × 4) = log(12)

Quotient Law

log(a/b) = log(a) − log(b)

Logs turn division into subtraction.

Example: log(50) − log(5) = log(50/5) = log(10) = 1

Power Law

log(aⁿ) = n · log(a)

Logs bring exponents down in front as a multiplier.

Example: log(8) = log(2³) = 3 · log(2)

These laws work for any consistent base, as long as you use the same base throughout the expression.

Evaluating Logarithms Without a Calculator

To evaluate log₍b₎(x) without a calculator, rewrite it as an exponent equation and solve.

Example 1: Find log₃(81)

Let log₃(81) = y. Then 3ʸ = 81. Since 3⁴ = 81, y = 4.

Example 2: Find log₄(1/16)

Let log₄(1/16) = y. Then 4ʸ = 1/16. Since 4⁻² = 1/16, y = −2.

Example 3: Find log₅(1)

Since any base raised to 0 equals 1: log₅(1) = 0.

Solving Logarithmic Equations

Type 1 — Single log equals a number

Rewrite as an exponent equation.

log₂(x) = 5 → x = 2⁵ = 32

Type 2 — Two logs equal each other

If log₍b₎(A) = log₍b₎(B), then A = B (same base, same result means same argument).

log(x + 1) = log(4) → x + 1 = 4 → x = 3

Type 3 — Use the laws to combine logs first, then solve

log(x) + log(3) = log(12)
log(3x) = log(12)
3x = 12
x = 4

Always check your solution: the argument of a log must be positive. A negative or zero argument is undefined.

The Change of Base Formula

Calculators typically only have log₁₀ and ln buttons. To evaluate log₍b₎(x) for any other base, use:

log₍b₎(x) = log(x) / log(b)

(or equivalently, ln(x) / ln(b))

Example: Find log₅(200)

log₅(200) = log(200) / log(5) ≈ 2.301 / 0.699 ≈ 3.30

Real-World Applications

Logarithms compress very large and very small numbers into manageable scales. They appear across science, technology, and everyday life.

pH scale — measures acidity. pH = −log₁₀([H⁺]). Each unit change represents a 10-fold change in acidity.

Richter scale — measures earthquake magnitude. A magnitude 7 earthquake releases about 31.6 times more energy than magnitude 6 (the scale is logarithmic).

Decibels — measures sound intensity. Every 10 dB increase represents a 10-fold increase in intensity, which is perceived as roughly twice as loud.

Compound interest and population growth — logarithms appear when solving for time in exponential growth equations such as A = Peʳᵗ (take the natural log of both sides to isolate t).

Computing — binary search runs in O(log₂ n) time; the height of a balanced binary tree is log₂(n) levels.

Common Mistakes

• Thinking log(a + b) = log(a) + log(b) — this is false. The product law covers log(ab), not log(a + b).
• Forgetting the argument must be positive — log(0) and log(−3) are undefined.
• Mixing up log and ln — they share the same laws but have different bases (10 vs e). Never substitute one for the other without adjusting.
• Confusing log₍b₎(b) with log₍b₎(1) — log₍b₎(b) = 1 and log₍b₎(1) = 0. A common slip is reversing these.
• Forgetting to check solutions — after solving a log equation algebraically, always verify the argument of every log is positive in the original equation.

Summary of Key Facts

Expression	Value	Reason
log₍b₎(1)	0	b⁰ = 1
log₍b₎(b)	1	b¹ = b
log₍b₎(bⁿ)	n	definition of log
b^(log₍b₎(x))	x	log and exponent cancel

Logarithms are one of the most practical tools in secondary mathematics — once you see how they reverse exponentiation, the laws follow naturally from the laws of indices you already know.