Learn With Examples
Earthquakes, sound, acidity, star brightness — whenever the world hands us numbers that span a billion-to-one range, we reach for the same trick. Logarithms are that trick, and they are far less abstract than school made them look.
In March 2011, an earthquake off the coast of Japan registered magnitude 9.1. Four years later, one in Nepal registered 7.8. On paper that looks like a modest difference — one and a bit points on a ten-point scale, the sort of gap you’d shrug at in an exam grade.
It isn’t. The Japanese earthquake released roughly ninety times more energy. Not ninety percent more. Ninety times.
That gap between how the number looks and what it means exists because the magnitude scale is logarithmic. And once you notice one logarithmic scale, you start seeing them everywhere: the decibels on your headphone warning, the pH on a bottle of cleaner, the f-stops on a camera, the octaves on a piano. All the same idea, all for the same reason.
This article explains what a logarithm actually is in plain language, then walks through the three scales people meet most — earthquakes, sound and pH — with real numbers you can check.
A logarithm answers “how many times did I multiply?”
Multiplication asks: I have 10, multiplied by itself 3 times — what do I get? Answer: 1,000.
A logarithm asks the reverse: I have 1,000 — how many 10s did I multiply to get here? Answer: 3. Written log₁₀(1000) = 3.
That’s it. A logarithm is a counter of multiplications, the way division is a counter of subtractions. Everything else follows.
Start with the powers, not the logs
Logarithms feel strange only because they’re usually taught before the thing they undo. So take the multiplication first:
Now read the same line backwards. Given 10,000, how many steps from 1? Four. So log₁₀(10,000) = 4. Given 100? Two steps, so the log is 2. The logarithm is simply the exponent, extracted and looked at on its own.
Here’s a shortcut that makes base-10 logs concrete forever: for a whole power of ten, the log is the number of zeros. A million has six zeros, so its log is 6. And for anything in between, the log sits between the neighbouring whole numbers — 5,000 is between 1,000 and 10,000, so its log is between 3 and 4 (about 3.7).
Look at the right-hand column. The values explode — 1 to a million — while the logs plod along, 0 to 6. That compression is the entire practical purpose of the tool. A quantity that spans a million-to-one range becomes a scale from 0 to 6, which fits on a chart, on a dial, and in a human head.
Which base? Base 10 is standard for measuring scales, because our number system is decimal. Base 2 appears throughout computing (each step is a doubling — that’s why log₂ shows up in algorithm analysis and in bits of information). Base e ≈ 2.718, the “natural log”, is standard in maths and physics because it makes the calculus clean. Same idea, different step size.
Example 1 — Earthquakes
Earthquake magnitude is the scale most people have heard of and almost nobody reads correctly. The number describes the amplitude of ground motion recorded by instruments, on a base-10 logarithmic scale.
So a magnitude 6 shakes the ground ten times as hard as a magnitude 5, and releases about 32 times more energy. Two steps up, from 5 to 7, means 100 times the shaking and about 1,000 times the energy.
What one magnitude step really means
tap a magnitudeMagnitude 4.0 Minor tremor — the kind most seismic regions get weekly
Felt indoors, rattling windows. Thousands occur every year worldwide.
Magnitude 5.5 Comparable to many moderate regional quakes
Furniture moves, weak buildings crack. Locally alarming, rarely deadly.
Magnitude 6.7 Northridge, California, 1994
Serious damage in built-up areas. The 1994 Northridge earthquake in California measured 6.7 and caused tens of billions of dollars in damage.
Magnitude 7.8 Nepal 2015 · Turkey–Syria 2023
Catastrophic across a wide region. The 2015 Nepal earthquake and the 2023 Turkey–Syria earthquake were both around 7.8.
Magnitude 9.1 Tōhoku, Japan, 2011
Among the largest ever recorded. The 2011 Tōhoku earthquake off Japan measured about 9.0–9.1 and triggered the tsunami that struck Fukushima.
Shaking multiplies by 10 per whole step; energy multiplies by about 31.6, because energy scales with the 1.5 power of magnitude. That is why a 9.1 is not “slightly worse” than a 7.8 — it releases roughly 90 times more energy.
This is why news coverage that says an earthquake was “upgraded from 7.5 to 7.8” is reporting something significant, not a rounding correction. That 0.3 is roughly a tripling of released energy.
It also explains why the scale has no upper limit but rarely exceeds 9.5. The magnitude depends on how much fault surface ruptures, and the planet simply doesn’t contain faults long enough to go much higher. The largest ever instrumentally recorded, in Chile in 1960, was about 9.5.
Example 2 — Sound and decibels
Human hearing is astonishing: the quietest audible sound and the threshold of pain differ in intensity by a factor of about one trillion. Writing that on a volume dial is hopeless. So sound is measured in decibels — a logarithmic scale that turns 1,000,000,000,000 into a tidy 0 to 120.
| Sound | Decibels | Intensity vs a whisper | Safe exposure |
|---|---|---|---|
| Rustling leaves | 20 dB | 0.1× | Unlimited |
| Whisper, quiet library | 30 dB | 1× | Unlimited |
| Normal conversation | 60 dB | 1,000× | Unlimited |
| Busy city traffic | 85 dB | 316,000× | About 8 hours |
| Motorbike, food blender | 94 dB | 2.5 million× | About 1 hour |
| Rock concert, chainsaw | 110 dB | 100 million× | Around 2 minutes |
| Jet engine at 30 m | 140 dB | 100 billion× | Immediate damage |
The third column is the part worth staring at. A rock concert isn’t “a bit louder” than conversation — it delivers roughly a hundred thousand times the sound intensity to your ear. The scale is compressing that so hard it looks almost reasonable.
The safe-exposure column follows from the same maths, and it’s the practical payoff. Hearing-protection guidance commonly halves the permitted exposure time for every 3 dB increase, because 3 dB is a doubling of intensity. That gives roughly 8 hours at 85 dB, 4 hours at 88, 2 hours at 91, 1 hour at 94. A small number on the dial is a big change in your ear.
Why your volume slider feels wrong. Going from 20% to 40% doesn’t sound twice as loud, because perception is itself roughly logarithmic. Doubling the electrical power adds only 3 dB, and 3 dB is barely noticeable. This is also why a 100-watt speaker isn’t twice as loud as a 50-watt one — it’s about 3 dB louder, which most people would describe as “slightly”.
Example 3 — pH and acidity
pH measures how many hydrogen ions are floating in a liquid. Those concentrations vary across an enormous range, so chemists did what physicists did with sound: took the logarithm. With one twist — pH uses the negative log, so that acidic things get small numbers.
| Substance | pH | Hydrogen ions vs pure water |
|---|---|---|
| Battery acid | 0 | 10,000,000× more |
| Lemon juice | 2 | 100,000× more |
| Cola | 2.5 | ~32,000× more |
| Black coffee | 5 | 100× more |
| Pure water | 7 | baseline |
| Human blood | 7.4 | ~2.5× less |
| Household bleach | 13 | 1,000,000× less |
Now the medical detail that makes this concrete. Human blood is held between about 7.35 and 7.45. Outside roughly 6.8 to 7.8, cells stop functioning and the situation is life-threatening. That sounds like a generous margin until you translate it: the difference between pH 7.4 and pH 6.8 is a four-fold increase in hydrogen ions. Your body defends that range so fiercely precisely because a small pH move is a large chemical one.
The same arithmetic explains why ocean acidification is taken seriously despite unimpressive-looking numbers. Surface ocean pH has fallen from roughly 8.2 to about 8.1 since the industrial era. A tenth of a point — and roughly a 25 to 30 percent increase in hydrogen ion concentration, because 100.1 ≈ 1.26.
On a logarithmic scale, small differences in the number are never small differences in the world. That is the whole reason the scale exists.
The same trick, four more places
Star brightness
Magnitude runs backwards — brighter stars get smaller numbers — and 5 steps equals exactly 100× the brightness, so one step is about 2.512×.
Camera f-stops
Each stop halves or doubles the light. Shutter speeds do the same. Photography is a base-2 logarithmic system with a friendlier name.
Music
An octave doubles the frequency, and every octave is divided into 12 equal ratio steps. Pitch perception is logarithmic, which is why the frets on a guitar get closer together.
Computing
Binary search takes log₂ n steps because it halves the problem each time. A billion items, about 30 steps.
The property that built the modern world
Before calculators, logarithms were not a topic — they were a labour-saving device, and an enormous one. The reason is this identity:
Look up two logs in a table, add them, look the answer back up, and you have multiplied two large numbers without multiplying anything. Navigators, astronomers and engineers used printed log tables and slide rules on exactly this principle for over three centuries, and the Apollo programme was still flying with slide rules in engineers’ pockets.
The same identity is why logarithmic charts are so useful today: on a log axis, anything growing by a constant percentage plots as a straight line. Compound interest, population growth, viral spread — all straight lines on log paper, and their slope tells you the growth rate directly.
The catch with log charts. That same compression can mislead. A curve that is exploding upward looks calm and nearly flat on a logarithmic axis. During the early COVID-19 period, the same case data plotted linearly and logarithmically produced very different emotional reactions from the same facts. Always check which axis you’re reading.
The three rules worth actually remembering
| Rule | What it does | Where you meet it |
|---|---|---|
| log(ab) = log a + log b | Turns multiplying into adding | Slide rules, combining decibel sources |
| log(a/b) = log a − log b | Turns dividing into subtracting | Ratios, dB comparisons between two signals |
| log(an) = n × log a | Pulls an exponent down into a multiplier | Solving for time in growth and decay problems |
That third rule is the one that earns its keep in real problems. Suppose an investment grows 7% a year and you want to know when it doubles. You need to solve 1.07t = 2 — and t is stuck up in the exponent where algebra can’t reach it. Take the log of both sides, and the rule pulls it down: t × log(1.07) = log(2), so t = log(2) / log(1.07) ≈ 10.2 years.
Incidentally, that’s where the “rule of 72” comes from — 72 divided by 7 is about 10.3. The mental shortcut is a logarithm in disguise.
Why we bother: three reasons
Compression
A trillion-to-one range becomes 0 to 120. Any scale spanning many orders of magnitude gets a logarithm sooner or later.
Matching perception
Human senses respond to ratios, not differences. Sound, brightness and pitch all work multiplicatively, so a log scale matches how we actually experience them.
Solving for exponents
Whenever the unknown is an exponent — doubling time, half-life, decay — the logarithm is the tool that gets it back down.
Check yourself
Five questions. Open each to check — the correct option is marked.
1. What is log₁₀(100,000)?
- 4
- 5
- 10
- 100
Count the zeros: five. So you multiply by ten five times to get from 1 to 100,000.
2. How much more energy does a magnitude 7 earthquake release than a magnitude 5?
- 2 times
- 100 times
- About 1,000 times
- 40 times
Two steps up. Shaking multiplies by 10 twice (100×), and energy by about 31.6 twice — roughly 1,000×.
3. A sound goes from 60 dB to 90 dB. How much has the intensity increased?
- 1.5 times
- 30 times
- 1,000 times
- 90 times
Every 10 dB is a factor of ten, and that’s three lots of 10 dB — so 10 × 10 × 10. To your ears it sounds roughly eight times louder, because perception compresses it again.
4. Lemon juice is pH 2, black coffee is pH 5. How much more acidic is the lemon juice?
- 2.5 times
- 3 times
- 1,000 times
- 30 times
Three whole steps down the pH scale, each a factor of ten. Lemon juice has about a thousand times the hydrogen ion concentration.
5. Why does adding logarithms multiply the original numbers?
- It’s a coincidence of base 10
- Logs are exponents, and multiplying powers means adding exponents
- Because logs are always whole numbers
- It only works for small numbers
10² × 10³ = 10⁵. The exponents add, and a logarithm is the exponent — which is exactly what made slide rules work.
Frequently asked questions
What is a logarithm in simple terms?
It’s the answer to “how many times do I multiply this base by itself to reach that number?” Since 10 × 10 × 10 = 1,000, the base-10 logarithm of 1,000 is 3. It’s the inverse of raising to a power, the way subtraction is the inverse of addition.
Why are earthquake and sound scales logarithmic?
Because the underlying quantities span enormous ranges — sound intensity varies about a trillion-fold across human hearing. A logarithmic scale compresses that into a usable 0 to 120, and it also matches how our senses respond, which is to ratios rather than absolute differences.
What’s the difference between log, ln and log₂?
Only the base. Written plainly, log usually means base 10 and each step multiplies by ten. ln is the natural log, base e ≈ 2.718, standard in calculus and continuous growth. log₂ is base 2, where each step is a doubling — the everyday base in computing.
Where are logarithms used in real life?
Earthquake magnitude, decibels, pH, star magnitude, camera f-stops, musical pitch, the Big O analysis of algorithms, half-life calculations, compound interest, information measured in bits, and every chart with a logarithmic axis.
Can you take the log of zero or a negative number?
No, not with real numbers. No power of 10 produces zero or a negative result — you can get closer and closer to zero (10⁻⁶ is 0.000001) but never reach it. That’s why log curves shoot downward without limit as they approach zero on the left.
The takeaway
A logarithm counts multiplications. That single sentence covers everything above: the magnitude scale counts factors of ten in ground motion, decibels count factors of ten in sound intensity, pH counts factors of ten in hydrogen ions, and an octave counts doublings of frequency.
The practical habit is to translate before reacting. When you see a jump of one on a logarithmic scale, silently multiply by ten instead of adding one. Magnitude 8 versus 7 is not “a bit worse”. pH 5 versus 6 is not “slightly more acidic”. 100 dB versus 90 dB is not “a touch louder”.
Try it on the nearest example to hand: check the pH printed on a bottle in your kitchen, or the decibel figure in your phone’s hearing-safety settings. Convert the number into a multiplication, and you’ll find the reading changes what it means to you. That’s the whole skill.
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