📖 What is a Logarithm?

A logarithm answers: "What power must I raise the base to, to get this number?" It is the inverse of indices.

Inverse of indices log_base(answer) = power Decibels · pH · Signal gain

🎮 Base · Power · Answer Explorer

ℹ️ How to use: Click a Base button and a Power button below. The index form and logarithm form both update instantly so you can see they describe the same fact.

Base:

Power:

same fact — two forms

      2^3 = 8  ↔  log₂(8) = 3 — "What power gives me 8 from base 2? Answer: 3"
    

🎮 Base · Power · Answer Explorer

ℹ️ How to use: Click a button for the base and a button for the power. Watch both the index form and logarithm form update live to show the same mathematical fact two ways.

Base:

2

Power:

3

2³ = 8
↔️ log₂(8) = 3

2³ = 8 ↔️ log₂(8) = 3 — "To get 8 from base 2, I need power 3"

Key facts

log_a(a) = 1 (a¹ = a)

log_a(1) = 0 (a⁰ = 1)

⚠ log(0) = ERROR | log(negative) = ERROR

💡 log button = log₁₀ | ln button = log_e

Evaluate examples from your notes

(a) log₃(9)

3^? = 9:3² = 9

Answer:log₃(9) = 2 ✓

(b) log₃(1/27)

Write:1/27 = 3⁻³

Answer:log₃(1/27) = −3 ✓

📐

Laws of Logarithms

Four laws derived from index laws. Learn them, then use the interactive matching game.

🔗 Rules Map — Tap an expression, then its simplified result

The law used is shown when you get it right!

Matched: 0 / 8

All four laws

Law 1: log(xy) = log x + log y

Law 2: log(x/y) = log x − log y

Law 3: log(xᵐ) = m · log x

logₐ(1) = 0 | logₐ(a) = 1

⚠ log(0) = ERROR always!

Worked examples

5 log 2 + 2 log 5

Law 3:log 32 + log 25

Law 1:log 800 ✓

log 10 + 2 log 3 − log 2

Law 4b:log 10 = 1

Laws 2+3:1 + log 9 − log 2 = log(10 × 4.5)

Answer:log 45 ✓

📈

Log Graphs

All log graphs pass through (1,0). Drag the slider to see how changing the base reshapes the curve.

y = log_a(x): passes through (1, 0) | passes through (a, 1) | defined x>0 only | asymptote x=0

Base: 2.0

      y = log₂(x)  |  passes through (1, 0) and (2, 1)  |  log₂(4) = 2
    

🟡 fixed point at (1, 0) · 🟢 point at (base, 1)

Key points: log₂(x)

log₂(½) = −1 (2⁻¹ = 0.5)

log₂(1) = 0 (2⁰ = 1)

log₂(2) = 1 (2¹ = 2)

log₂(4) = 2 (2² = 4)

log₂(8) = 3 (2³ = 8)

Log is the inverse of exponential

The log graph is the reflection of y=a^x in the line y=x. They undo each other:

ln(e^x) = x and e^{ln x} = x

log(10^x) = x and 10^{log x} = x

💡 Use this to solve equations where x is in a power.

✏️

Worked Examples

Applying the laws step by step. Always state which law you use.

Example 1 — Evaluate log₂(512)

Ask:2^? = 512 → 2,4,8,16,32,64,128,256,512

Answer:log₂(512) = 9 ✓

Example 2 — Evaluate log₈(1/64)

Write:1/64 = 64⁻¹ = (8²)⁻¹ = 8⁻²

Answer:log₈(1/64) = −2 ✓

Example 3 — Simplify: 2 log 3 − 3 log 2

Law 3:log 9 − log 8

Law 2:log(9/8) ✓

Example 4 — Simplify: 5 log 2 + 4 log 3 − 3 log 4

Law 3:log 32 + log 81 − log 64

Laws 1+2:log(32×81/64)

Answer:log(81/2) ✓

Example 5 — Simplify: log 10 + 2 log 3 − log 2

Law 4b:log 10 = 1

Laws 2+3:1 + log 9 − log 2 = log 10 + log(9/2)

Answer:log 45 ✓

Example 6 — Simplify: log 12 − 2 log 2 + log 3

Law 3:log 12 − log 4 + log 3

Laws 1+2:log(12/4 × 3) = log(3×3)

Answer:log 9 ✓

🔧

Solving Equations Using Logs

When x is in the power, take logs of both sides and use Law 3 to bring it down.

Key method: a^x = b ⇒ x log a = log b ⇒ x = log b / log a

4-step method

1Take log of both sides
2Apply Law 3 — bring power to front
3Rearrange to isolate x
4Calculator for numerical answer

Worked: solve 3^x = 5

Take log:log 3^x = log 5

Law 3:x log 3 = log 5

Rearrange:x = log 5 / log 3

Answer:x = 1.465 ✓

Harder: solve 3^x = 5^x−2

Take log:x log 3 = (x−2) log 5

Expand:x log 3 = x log 5 − 2 log 5

Rearrange:2 log 5 = x(log 5 − log 3) = x log(5/3)

Answer:x = 2 log 5 / log(5/3) ✓

Solve 4^x = 12

x log 4 = log 12x = 1.792 ✓

Solve e^x = 8

Take ln:x = ln 8 = 2.079 ✓

💡 For base e: use ln directly.

🌍

Logarithms in Real Life & Engineering

Logarithms appear wherever quantities span enormous ranges — sound, earthquakes, signals, chemistry, finance and more.

🔊

Sound & Acoustics

Decibel Scale

dB = 10 log(I / I₀)

A sound 10× more intense is only +10 dB louder because your ear responds logarithmically. A jet engine (140 dB) is 10¹⁴ times more intense than the quietest sound you can hear.

⚗️

Chemistry & pH

Acidity Scale

pH = − log[H⁺]

Each pH unit = 10× more or less acidic. Stomach acid pH 1, pure water pH 7, bleach pH 13. The log scale makes the enormous range of concentrations manageable.

📡

Electronics & Signal Gain

Amplifiers & Filters

Gain (dB) = 20 log(V₀ᵘᵗ / Vᵖₙ)

Engineers use logs to compare signal voltages without needing huge numbers. A gain of 40 dB means the output is 100× the input voltage.

☢️

Radioactive Decay

Half-life & Carbon Dating

t = − ln(N / N₀) / λ

N = N₀ e^−λt — to find when something decayed, take ln of both sides. Used in nuclear engineering and carbon-14 dating of ancient objects.

🌍

Earthquakes

Richter Scale

M = log(A / A₀)

Each step up the Richter scale = 10× more ground motion. A magnitude 7 quake is 1 000× stronger than magnitude 4 — but only 3 steps on the scale.

💰

Finance & Growth

Doubling Time & Compound Interest

t = ln(2) / r

At 7% annual growth: t = 0.693 / 0.07 ≈ 10 years to double. Logs turn exponential growth into a simple linear equation.

💡 The pattern: whenever a quantity can vary by millions or billions, engineers use a logarithmic scale to keep the numbers human-sized.

🧮

Interactive Logarithm Solver

Evaluate any logarithm, solve log equations, and apply the log laws step by step. Change the values and the working updates instantly.

📊 Log Evaluator — log_b(x) = ?Step-by-step

ℹ️ How to use: Type a base and a number into the boxes. The tool calculates log_b(x) and shows the connection back to index form.

log

Base b

(

Number x

)

Enter a base and number to evaluate.

👆 Try: log₂(8) · log₁₀(1000) · log₃(81) · log₅(0.04) · ln(e)

📐 Log Laws ExplorerLaws 1–4 with working

ℹ️ How to use: Enter two numbers A and B and a base, then use the dropdown to pick a log law. The full step-by-step simplification is shown.

Base b

A

B

Choose a law

Select a law from the dropdown to see the working.

👆 Try: log(4)+log(25) · log(100)−log(20) · 3 log(2) · log₂(2)

The 4 Laws at a Glance

Law 1 log(AB) = log A + log B

Law 2 log(A/B) = log A − log B

Law 3 log(Aⁿ) = n log A

Law 4a logₐ(a) = 1

Law 4b logₐ(1) = 0

Common mistakes

⚠ log(A+B) ≠ log A + log B

⚠ log(A×B) ≠ log A × log B

⚠ log(0) and log(negative) are undefined

✅ log without a base shown means log₁₀ on most calculators. ln means log𝑒 (natural log).

🎮

Games

Log Blitz to drill values, plus a Laws Match game.

⚡ LOG BLITZ — Evaluate the Logarithm

ℹ️ How to play: Click an expression on the left, then click its matching simplified form on the right. Correct pairs disappear — aim to match all 8 without mistakes.

Score: 0 Streak: 0 🔥 Q 1/10

log₂(8)

🔗 LAWS MATCH — Expression to Simplified Form

Tap left, then right

📝

Practice Questions

Show all working. State which law you use.

🟢 Tier 1 — Evaluate

ℹ️ How to use: Work through each question and write down your answer. When ready, click Show Answer to reveal the full worked solution and mark scheme.

Q1

Evaluate: (a) log₃(9) (b) log₂(32) (c) log₅(125) (d) log₁₀(10000)

[4]

(a) 2 (b) 5 (c) 3 (d) 4

Q2

Evaluate: (a) log₄(64) (b) log₂₅(5) (c) log₃(1/27) (d) log₈(2)

[4]

(a) 3 (b) 1/2 (c) −3 (d) 1/3

🟡 Tier 2 — Apply laws

Q3

Write as log N: (a) log 3 + log 5 (b) log 16 − log 2 (c) 3 log 4 (d) 2 log 3 − 3 log 2

[4]

(a) log 15 (b) log 8 (c) log 64 (d) log(9/8)

Q4

Simplify: (a) 5 log 2 + 2 log 5 (b) log 128 − 7 log 2 (c) log 10 + 2 log 3 − log 2

[3]

(a) log 800 (b) 0 (log 128 = 7 log 2) (c) log 45

🔴 Tier 3 — Solve equations

Q5

Solve: (a) 10^x = 5 (b) e^x = 8 (c) 4^x = 12 (d) 3^x = 2

[4]

(a) 0.699 (b) 2.079 (c) 1.792 (d) 0.631

Q6

Solve 3^x = 5^x−2. Give answer to 3 d.p.

[5]

Hint: take log both sides, Law 3, rearrange, factorise

x log 3 = (x−2) log 5 ⇒ 2 log 5 = x log(5/3)
x = 2 log 5 / log(5/3) ≈ 4.301 ✓

🎯 Score: Q1–2 (8) · Q3–4 (7) · Q5–6 (9)

⏱ Timed Quiz — 15 seconds per question!

ℹ️ Timed Quiz: Each question gives you 15 seconds. Click the correct answer before time runs out — the timer ring shrinks as your time decreases. Your score and percentage are shown at the end.