log · Logarithm
The language that compresses vast differences — the logarithm.
How can you fairly compare numbers as absurdly far apart as 1 and 10 billion, on a single screen?
Read the conditions on the base and the argument.
The invention said to have doubled the life of the astronomer.
Open the immersive version →A reading path containing this lesson: Reverse repeated multiplication · 3/3
Why does this question matter?
Earthquake magnitude, sound level (decibels), star brightness, stock returns — real-world values often differ by factors of 10, 100, or 1000.
Plot those values directly and the small ones get crushed flat. You can see the gap between 1 and 10 fine, but the gap between 10,000 and 100,000 barely fits on screen.
A logarithm turns the multiplicative question 'how many times bigger?' into the additive question 'how many multiplications?'. That's what lets even astronomically large values sit on a small, readable scale.
Pause and try it yourself
Make a prediction first. Change one input at a time and compare what happens with what you expected.
| The real number (it explodes) | Times multiplied by 10 |
|---|---|
| 1 (never multiplied — the floor) | 0 |
The ladder now = the formula now
Press ×10 to stack one rung on the ladder.
Both lines change together every time you press ×10.
Read the idea from another angle
A logarithm turns the world of multiplication into the world of addition.
The multiplicative relationship 10 × 10 × 10 = 1000 becomes the additive relationship 1 + 1 + 1 = 3 in the world of logs. Logarithms were born to turn hard multiplication into easy addition.
A sense for 'how many digits', not the exact number.
The integer part of log10(x) is essentially the digit count of x minus one. Reading a log value trains you to judge 'roughly what scale is this?' before you need the exact figure.
Now read the definitions with their conditions
The pattern you found in the experiment — 'the log grows by exactly 1 every time the value multiplies by 10' — is exactly the definition and properties of the logarithm.
Definition of a logarithm
A logarithm answers 'how many times must I multiply 10 to get x?'. As you saw on the slider, when x = 1000 the answer is 3.
Multiplication → addition
The log of a product equals the sum of the logs. This is the precise statement of 'logs turn multiplication into addition'.
Division → subtraction
By the same logic, division becomes subtraction in the world of logs.
Power → multiplication
Multiplying a by itself n times becomes, in the world of logs, multiplying log a by n. A complicated power collapses into a simple product.
Make one more distinction
If the log value doubles, the original number roughly doubles too.
When log goes from 2 to 4, the original number goes from 100 to 10,000 — a 100x jump. One tick on the log scale isn't addition, it's one multiplication.
Logs are a calculator thing, disconnected from real life.
Earthquake magnitude, loudness (dB), acidity (pH), star brightness — most of the units our senses use are already logarithmic, because our senses themselves respond logarithmically.
Where else can this idea help?
Earthquake magnitude (Richter scale)
A magnitude-7 quake releases about 32 times more energy than a magnitude-6. Because the log scale compresses that multiplicative gap, we can talk about it as a difference of just 1.
Loudness (decibels)
For a hypothetical 90 dB level difference, the pressure ratio is 10^(90/20), about 31,623. In the same medium, the corresponding intensity ratio is 10^(90/10), one billion. Distinguish pressure from intensity before choosing the factor 20 or 10.
Acidity (pH)
Every drop of 1 in pH means the hydrogen ion concentration gets 10 times stronger. That's how pH 3 vinegar ends up 10,000 times more acidic than pH 7 water, expressed in a single digit.
Long-run stock charts
A 10-plus-year stock chart needs a log scale to fairly compare early gains with recent ones — because the same slope means the same rate of return.
Explain it in your own words
Try another question
Q1Without a calculator: roughly what is log₁₀(50,000)? (a range is fine, not the exact value)Show answer ▾
It's between 10⁴ = 10,000 and 10⁵ = 100,000, so the answer is between 4 and 5 (actually about 4.7). A sense of 'digit count minus one, up to digit count' is enough — a logarithm is a tool for reading scale.
Q2City A has 30,000 people, city B has 30 million. How many rungs apart are they on a log scale? (1 rung = ×10)Show answer ▾
30 million ÷ 30 thousand = 1,000 = 10³, so exactly 3 rungs. Even though the raw numbers are wildly different, the log scale lets you place them side by side.
Q3If sound intensity becomes 10 times greater, how many decibels does it gain?Show answer ▾
Exactly +10dB. Decibels are defined as 10 × log₁₀(intensity ratio). At 100x it's +20dB — multiplication becoming addition, exactly as logs promise.
Try answering yourself before revealing it — getting it wrong is where learning starts.
What question comes next?
A reading path containing this lesson: Reverse repeated multiplication · 3/3
Previous concept
Functions
A logarithm is, in the end, the function y = log x. The function intuition of 'a fixed input decides a fixed output' is the foundation for reading logs.
← Functions labLeads to next
Derivatives
Once you're comfortable using logs to handle huge values, it's time to look at how fast a value is changing. A derivative captures the slope at a single instant — on a curve that bends gently, just like a log graph.
Go to the Derivatives lab →Related lab
Compound Interest
'How many years to double?' is fundamentally a log question. See the Rule of 72 for what it really is.
Go to the Compound Interest lab →Related lab
Logarithmic Functions
Unroll a logarithm into a function and you get the log scale — the skill that keeps you from being fooled by a news graph.
Go to the Logarithmic Functions lab →