Properties of Logarithms — Explained, With a Free Practice Game 7+

What is a logarithm?

Take the number 100 and ask: how many times do I multiply 10 to get there? Twice — 10 · 10 = 100. That is the logarithm: log₁₀100 = 2.

The number being multiplied is the base, written small and low. A base can be any positive number except 1 (multiplying 1 by itself gets you nowhere, so the question would have no answer). The two you meet most often are 10 and 2:

• log₁₀10 = 1 — ten taken once;
• log₁₀100 = 2, log₁₀1000 = 3, log₁₀1,000,000 = 6 — count the zeros;
• log₂8 = 3 — two multiplied three times: 2 · 2 · 2 = 8;
• log₂1024 = 10 — ten doublings land you near a thousand.

Logarithms and exponents are the same sentence read from opposite ends. 10³ = 1000 says "here is what you get after multiplying ten three times." log₁₀1000 = 3 says "here is how many times you had to multiply to reach a thousand." That is why the logarithm is called the inverse of exponentiation.

One habit saves a lot of trouble later: you can only take the logarithm of a positive number. No amount of multiplying a positive base produces zero or a negative result, so "how many times to reach −5" simply has no answer.

The properties of logarithms

Throughout, a is the base (a > 0, a ≠ 1) and x, y are positive numbers.

• Product rule: log_a(x · y) = log_ax + log_ay. Multiplication outside becomes addition inside.
• Quotient rule: log_a(x / y) = log_ax − log_ay. Division becomes subtraction.
• Power rule: log_a(xⁿ) = n · log_ax. The exponent moves out in front as a multiplier.
• Two anchor values: log_aa = 1 and log_a1 = 0. The base taken once gives one; taken zero times gives zero.
• Change of base: log_ax = log_bx / log_ba. You need it when your calculator only offers log (base 10) and ln (base e).

There is also the identity a^log_ax = x, which is just the definition said out loud: take the base as many times as the logarithm says, and you are back to the original number.

Why multiplication turns into addition

The product rule looks like a magic trick until you count zeros by hand. Multiply 100 by 1000:

• 100 has two zeros, so log₁₀100 = 2;
• 1000 has three zeros, so log₁₀1000 = 3;
• 100 · 1000 = 100,000 — five zeros, and 2 + 3 = 5.

Zeros stack when you multiply. A logarithm counts zeros, so logarithms add. Division mirrors it exactly: 100,000 / 100 = 1000, and 5 − 2 = 3.

The mistake almost everyone makes once

log(x + y) is not log x + log y. The rules cover multiplication and division only. Check it: log₁₀(10 + 90) = log₁₀100 = 2, while log₁₀10 + log₁₀90 is clearly more than 2. A sum inside a logarithm has to be worked out first, and only then does the logarithm get taken.

Worked examples: how to solve logarithms

1. Evaluate log₃81. Ask how many 3s multiply to 81 and count out loud: 3, 9, 27, 81 — four steps. Answer: 4.

2. Evaluate log₁₀0.001. Each division by ten is one step down: 1 → 0.1 → 0.01 → 0.001. Three steps, all downward. Answer: −3. A negative logarithm does not mean "no answer" — it means you divided instead of multiplying.

3. Simplify log₂8 + log₂4. By the product rule this is log₂(8 · 4) = log₂32 = 5. Check separately: 3 + 2 = 5.

4. Solve log₂x = 5. The logarithm says "two taken five times," so x = 2⁵ = 32.

5. Simplify log₅125². Bring the exponent out front: 2 · log₅125 = 2 · 3 = 6.

The move is always the same: rewrite the number inside as a power of the base, then read off the exponent.

Logarithmic scales in the real world

A logarithmic scale is a ruler whose steps are equal in length but mean multiplication rather than addition. On an ordinary ruler running to 100,000, the first few marks collapse into a dot. On a log scale, 1, 10, 100, 1000 and 100,000 sit evenly spaced — so an ant and a mountain fit on one drawing.

• Loudness in decibels. Adding 10 dB means ten times the sound energy, even though your ear registers something much milder.
• Earthquake magnitude. Neighbouring magnitudes differ by roughly ten times in the amplitude of ground shaking.
• Acidity, pH. Each pH unit is ten times more or less acid; the smaller the number, the more acidic the liquid.
• Star brightness. Astronomers use a logarithmic scale too — otherwise the Sun and a faint star could not share a chart.

In every case people chose to count "how many steps" instead of "how many times over," for the simple reason that steps are easier to add than factors are to multiply.

How the game trains these properties

The game above is not decoration for the text — it drills the exact moves described here. The child stands at a cannon, a star cruiser slides out of the clouds, and glowing rings turn around its shield: exactly as many rings as the shield number has zeros. Every pull of the lever adds one ring to the barrel, stretches the beam by one segment, and the sight jumps 10 → 100 → 1000. Equal steps, ten-times growth — a logarithmic scale built by hand.

The 100 levels follow the same sequence as this page:

• levels 1–25 — base-10 logarithms, i.e. counting zeros;
• 26–40 — shields that land between rungs, when the answer is not a whole number;
• 41–55 — powers of two: each click is now a doubling;
• 56–70 — twin guns, where two beams join end to end — that is log x + log y standing in for multiplication;
• 71–80 — the weakener divides the shield by ten, subtracting one step;
• 81–92 — the real-world scales: siren loudness, tremors, star brightness, acidity;
• 93–100 — a growing threat: how many steps are left before the red line.

The word "logarithm" is not spoken until level 20, by which point the child has been counting rings for weeks and the term simply names something familiar. There is no losing and no timer: a miss lights up the gap on the sky ladder and says how many times off the shot was.