Logarithms – The Inverse of Exponents
A logarithm answers one very practical question: “what power do I need to raise this base to, to get this number?” Written logb(x), it is the exact inverse of an exponential function – where bk = x asks “what do I get?”, logb(x) = k asks the reverse: “what power gave me this?” That single reversal turns out to be one of the most useful ideas in all of mathematics, because it takes numbers that grow explosively – earthquake energy, sound intensity, computing time, acidity – and compresses them down into a scale that's actually easy for humans to read, compare, and reason about.
Logarithms were invented in 1614 by the Scottish mathematician John Napier, two decades before the modern exponent notation even existed. Napier's original goal was entirely practical: astronomers and navigators of the time spent enormous amounts of time doing long multiplication and division by hand, and Napier realised that because logarithms turn multiplication into addition, his log tables could turn a slow multiplication into a fast lookup-and-add. The English mathematician Henry Briggs visited Napier in 1615 and, working together, they refined the idea into base-10 (“common”) logarithms, the version most useful for everyday calculation. For the next 350 years, until pocket calculators appeared in the 1970s, log tables and slide rules (mechanical devices built entirely around the logarithm's addition trick) were the standard tools of scientists and engineers – they were even used to calculate the trajectories for the Apollo Moon missions. The other essential logarithm, the natural logarithm (base e ≈ 2.71828, written ln(x)), grew out of Jacob Bernoulli's and Leonhard Euler's work on continuous growth, covered on the previous page.
What Does a Logarithm Mean?
logb(x) = k means exactly the same thing as bk = x. The logarithm is asking: “b to what power gives x?”
“2 to what power gives 8?” Since 23 = 8, log₂(8) = 3.
“10 to what power gives 1,000?” Since 103 = 1,000, log₁₀(1,000) = 3.
Common Log Bases
| Name | Base | Notation | Typically Used For |
|---|---|---|---|
| Common logarithm | 10 | log(x) or log₁₀(x) | Scientific scales (pH, decibels, Richter) |
| Natural logarithm | e ≈ 2.71828 | ln(x) | Calculus, continuous growth and decay |
| Binary logarithm | 2 | log₂(x) | Computer science, information theory |
The Laws of Logarithms
Product law: logb(mn) = logb(m) + logb(n) – this is the whole reason logarithms were invented: turning multiplication into addition.
Quotient law: logb(m/n) = logb(m) − logb(n)
Power law: logb(mp) = p · logb(m)
Change of base: logb(x) = log(x) ÷ log(b), useful for computing any base on a calculator that only has log or ln.
log₂(8) + log₂(4) = log₂(8 × 4) = log₂(32).
Since 25 = 32, the answer is 5 (matching 3 + 2 = 5 directly).
Solving Exponential Equations with Logarithms
Logarithms are the standard tool for solving an equation where the unknown is stuck in the exponent.
Taking log₂ of both sides: x = log₂(32). Since 25 = 32, x = 5.
Logarithms in Daily Life
Logarithmic scales exist because human senses and many real-world quantities respond to ratios, not fixed amounts – a phenomenon called the Weber–Fechner law. A sound twice as intense doesn't feel “twice as loud” to your ears; it feels like a fixed step louder. Logarithms are the natural language for measuring things that work this way.
| Where | How Logarithms Are Used |
|---|---|
| pH scale (chemistry) | pH = −log₁₀[H⁺]; each drop of 1 pH unit means 10 times more acidic |
| Richter / moment magnitude scale (earthquakes) | Each whole number increase means about 32 times more energy released |
| Decibels (sound intensity) | dB = 10 × log₁₀(intensity ratio); a 10 dB rise sounds roughly twice as loud |
| Computing & algorithms | Binary search and balanced trees run in O(log n) time – searching a billion items takes only about 30 steps |
| Music | Octaves and semitones divide pitch using a logarithmic (not linear) scale |
| Astronomy | Star brightness is measured on a logarithmic magnitude scale |
| Finance | The “Rule of 72” (time to double an investment ≈ 72 ÷ interest rate) is a shortcut derived from logarithms |
Each pH unit is a factor of 10, and the difference is 7 − 4 = 3 units.
103 = 1,000 times more acidic.
Each whole number is about 32 times more energy, and the difference is 2 units.
322 ≈ 1,000 times more energy.
Key Takeaways
- logb(x) = k means the same thing as bk = x – a logarithm is the inverse of an exponent.
- The product, quotient, and power laws turn multiplication, division, and powers into addition, subtraction, and multiplication.
- Logarithms solve equations where the unknown is in the exponent.
- Because many real quantities (sound, acidity, earthquakes, computing time) span an enormous range, they are measured on logarithmic scales that compress that range into something readable.