Logarithms, explained simply
What a logarithm is, why log base 10 and log base 2 both matter, the rules that make them useful, and where they turn up in everyday technology.
A logarithm is an exponent wearing a different hat. That is the whole idea, and it is usually buried under so much notation that people who would happily handle powers and roots freeze when they see log. Strip away the formality and a logarithm is just a question: what power do I raise this base to, to get this number? Everything else — the rules, the strange bases, the decibels and pH values on the news — follows from that one sentence.
A logarithm is a backward exponent
When you write 10³ = 1000, the base is 10, the exponent is 3, and the result is 1000. A logarithm asks the reverse question: given the base 10 and the result 1000, what was the exponent? The answer is 3, and it is written:
log₁₀(1000) = 3
Read it out loud as “log base ten of one thousand equals three”. Formally, logb(x) = y is exactly the same statement as by = x. The two forms carry identical information; the log form is simply solved for the exponent. A few more to fix the idea:
- log₂(64) = 6, because 2⁶ = 64.
- log₅(125) = 3, because 5³ = 125.
- log₁₀(0.01) = −2, because 10−2 = 0.01.
Two boundary cases fall out of the definition immediately. Any base raised to the power zero gives 1, so the log of 1 is always 0. And any base raised to the power one is itself, so logb(b) = 1. The logarithm calculator computes any base, reports the natural log and the base-10 log at the same time, and shows the antilog check — raising the base back to the computed exponent to confirm it returns the original number.
The three bases worth knowing
Logs work in any base greater than zero except 1, but three show up constantly.
- Base 10 (common logs). The default on most calculators. It matches our decimal number system, so it is used for pH, the decibel scale, and orders of magnitude. Written log or log₁₀.
- Base e ≈ 2.71828 (natural logs). The base of continuous growth and decay. Written ln. It appears in compound interest, population models, radioactive decay, and calculus, because e is the base for which the logarithm has the simplest rate of change.
- Base 2 (binary logs). The mathematics of doubling and halving. It governs data sizes, binary representations, and how many steps a search needs. Written log₂.
Because a calculator cannot have a button for every base, the change-of-base formula bridges them: logb(x) = ln(x) ÷ ln(b). The logarithm calculator uses exactly this identity, computing the natural logs and dividing. To find log₂(10), for example: ln(10) ÷ ln(2) = 2.302585 ÷ 0.693147 ≈ 3.3219.
The rules that make logarithms useful
Logs earn their place because they turn hard operations into easy ones. Multiplication becomes addition, division becomes subtraction, and powers become multiplication.
| Rule | Statement | What it does |
|---|---|---|
| Product | log(xy) = log(x) + log(y) | turns multiplication into addition |
| Quotient | log(x ÷ y) = log(x) − log(y) | turns division into subtraction |
| Power | log(xⁿ) = n · log(x) | pulls an exponent down as a multiplier |
| Identity | log₅(b) = 1, log₅(1) = 0 | defines the scale |
| Change of base | log₅(x) = ln(x) ÷ ln(b) | converts any base to one you have |
For example, log₁₀(200) = log₁₀(2 × 100) = log₁₀(2) + log₁₀(100) ≈ 0.3010 + 2 = 2.3010. Before electronic calculators, engineers multiplied large numbers by adding their logs on a slide rule and then looking up the antilog. The same property lets you solve equations where the unknown is an exponent: if 3x = 81, then x = log₃(81) = 4, because 81 is 3 to the fourth power.
Where logarithms appear in everyday technology
Logs are not an academic curiosity. They are unavoidable whenever a quantity ranges over many orders of magnitude, because they compress huge ranges into manageable numbers.
Decibels (sound and signal)
Sound and signal power are measured in decibels, defined as 10 · log₁₀(P ÷ P₀) for a power ratio. The log means the scale is multiplicative, not additive. Every increase of 10 dB is ten times the power. An increase of 3 dB is very close to a doubling, because 10 · log₁₀(2) ≈ 3.01. Going from 60 dB to 90 dB sounds like a modest change on paper, but it is 10⁵ = 1,000 times the power.
pH (acidity)
pH is the negative base-10 log of the hydrogen-ion concentration: pH = −log₁₀[H⁺]. A solution with [H⁺] = 10−5 mol/L has pH 5. Because it is a log scale, one pH unit is a tenfold change in concentration. A solution at pH 3 is 100 times more acidic than one at pH 5. This is why diluting acid to change the pH by one unit takes a large volume of water.
Earthquakes (magnitude scales)
Earthquake magnitude is logarithmic, so each whole number represents a tenfold increase in ground motion. The energy released rises even faster: roughly 101.5 ≈ 31.6 times per magnitude unit, which is the square root of 1,000. A magnitude 7 quake therefore releases about 1,000 times the energy of a magnitude 5, even though the numbers differ by only two. Without the log scale, the range from barely felt tremors to catastrophic quakes would need about eight decimal places to write down.
Data, bits, and binary
Base-2 logs count how many times you can double. One kibibyte is 2ⁱ⁰ = 1,024 bytes. A 10-bit value can represent 2ⁱ⁰ = 1,024 distinct states, and 20 bits gives 2²⁰ = 1,048,576. The number of bits needed to represent a number n is floor(log₂(n)) + 1; for 255 that is floor(7.994) + 1 = 8 bits, which is why a byte can hold 0 to 255. The same base-2 doubling underlies binary search: doubling the size of a sorted list adds only one comparison, so a list of a million items takes about 20 steps. The number base converter shows how binary, octal, decimal, and hexadecimal line up, and the 2⁴ relationship between bits and hex digits is a log fact.
Scientific notation
The exponent in scientific notation is the integer part of the base-10 log. The number 4.5 × 10−4 has a base-10 log of about −3.35, because log₁₀(4.5) ≈ 0.653 and 0.653 − 4 = −3.347. This is why multiplying numbers in scientific notation means multiplying the coefficients and adding the exponents — the product rule for logarithms in disguise. The scientific notation converter switches between the two forms, and the exponent it reports is the logarithm rounded down to a whole number.
Reading a logarithmic scale
A log scale is a number line where equal distances represent equal multiplications. On a base-10 log chart, the gap from 1 to 10 is the same width as the gap from 10 to 100 and from 100 to 1,000. That is why charts of viral spread or earthquake frequency use them: plotting a range from 1 to 10,000,000 on a linear axis renders everything below a million as a flat line against the bottom.
| Power of 10 | Value | log₁₀ |
|---|---|---|
| 10⁰ | 1 | 0 |
| 10¹ | 10 | 1 |
| 10² | 100 | 2 |
| 10³ | 1,000 | 3 |
| 10⁴ | 10,000 | 4 |
| 10⁻¹ | 0.1 | −1 |
| 10⁻² | 0.01 | −2 |
Common errors
- log(a + b) is not log(a) + log(b). The addition rule applies to products, not sums. There is no simple rule for the log of a sum.
- Logs of zero or negative numbers are undefined. No power of a positive base produces zero or a negative number. The calculator stops and says logs are only defined for positive numbers.
- Base 1 and non-positive bases are forbidden. 1 raised to any power is always 1, so the equation 1y = x has no solution unless x = 1, and then every y works. A base must be positive and not equal to 1.
- Confusing log and ln. ln uses base e, log usually means base 10. They are different numbers: ln(10) ≈ 2.3026, while log₁₀(10) = 1.
- Assuming a log is always small. log₁₀(1,000,000,000) is just 9, but that compression is the point, not a sign that the original number was small.
Frequently asked questions
What is the difference between log and ln?
Only the base. ln is the logarithm to base e ≈ 2.71828, used for continuous growth and calculus. log on most calculators and in most engineering contexts means base 10. The base is sometimes written explicitly as a subscript to remove doubt, as in log₂ or log₁₀.
Why can’t I take the logarithm of a negative number?
Because a positive base raised to any real power always produces a positive result. There is no exponent y for which 10y equals, say, −5. Complex numbers extend this, but for ordinary arithmetic the log of a negative number has no real answer.
What is the change-of-base formula used for?
It converts a log in an inconvenient base into two logs you can actually calculate, usually natural logs: logb(x) = ln(x) ÷ ln(b). It is how a calculator with only an ln button can return log₂, log₅, or any other base.
Are logarithms still useful now that calculators exist?
More than ever, just not for arithmetic shortcuts. They measure anything that spans many orders of magnitude — sound, acidity, earthquake energy, data, interest compounding — and they turn exponential relationships into straight lines so they can be analysed. The slide rule is gone; the underlying math is everywhere.