What Is a Logarithm?

Earthquake magnitudes, the loudness of sound in decibels, the pH of a swimming pool — all are measured on logarithmic scales, where each step up means ten times as much. Scientists reach for these scales because a logarithm neatly counts those tenfold jumps, taming numbers that would otherwise run from tiny to astronomical.

You already know how to grow a number by multiplying it by itself: 2, 4, 8, 16, 32, \dots — doubling again and again. A logarithm runs that process backwards and answers one question: what power? Handed the number 8 and told the base is 2, a logarithm asks: how many times must I multiply 2 by itself to reach 8?

Since 2^3 = 8, the answer is 3. We write that as \log_2(8) = 3. Read it aloud: "log, base two, of eight, is three" — or really, "the power that turns 2 into 8 is 3." A logarithm is nothing more mysterious than an exponent in disguise — an index, pulled out into the open.

These two lines say exactly the same thing, just written from two angles:

\log_b(x) = y \quad\Longleftrightarrow\quad b^{\,y} = x

The base b stays the base on both sides; the exponent y is the log's answer; and x is the number you fed in. Press play to watch 2^3 = 8 turn into \log_2(8) = 3.

A logarithm undoes a power

So a logarithm is the inverse of raising to a power, the way subtraction undoes addition or division undoes multiplication. Raise 2 to the power 3 and you get 8; take \log_2 of 8 and you get 3 straight back. They cancel each other out:

\log_b\!\left(b^{\,y}\right) = y \qquad\text{and}\qquad b^{\,\log_b(x)} = x

Whenever you can rewrite a number as a power of the base, you can read off its logarithm at a glance:

\log_2(8) = 3,\qquad \log_3(9) = 2,\qquad \log_{10}(1000) = 3

Two neat patterns fall straight out of the definition. Since b^1 = b, the log of the base itself is always 1; and since b^0 = 1, the log of 1 is always 0 — no matter what the base is:

\log_b(b) = 1,\qquad \log_b(1) = 0

Worked examples: just ask "what power?"

Every one of these is the same move. Name the base, name the target, and ask how many times the base multiplies up to the target.

Example 1 — \log_2(16). What power of 2 gives 16? Count the doublings: 2, 4, 8, 16 — that is 2^4. So \log_2(16) = 4.

Example 2 — \log_{10}(1000). What power of 10 gives 1000? Count the zeros: 1000 = 10^3. So \log_{10}(1000) = 3. (For base 10, the log is simply how many zeros.)

Example 3 — \log_3(27). What power of 3 gives 27? Since 3 \times 3 \times 3 = 27 = 3^3, we get \log_3(27) = 3.

Example 4 — going the other way. Suppose you are told \log_5(x) = 2. Rewrite it in index form: 5^2 = x, so x = 25. Whenever a log equation looks confusing, flip it into a power and it becomes obvious.

The two bases you'll meet everywhere

A logarithm can have any positive base, but two show up so often they get their own shorthand:

Base 10 — the "common" logarithm. Written \log(x) with no base shown (the button marked log on your calculator). It matches the way we write numbers in tens, so \log(1000) = 3.
Base e — the "natural" logarithm. Written \ln(x), where e \approx 2.718 is a special constant that runs through all of growth and decay in nature. It is the log you meet most in higher maths and science.

Logarithms make a handy ruler for things that grow by multiplying — they tame numbers that race off the top of the page, which is why they show up on exponential and real-life graphs. Coming up, they gain three powerful laws of logarithms.

Two traps snare almost everyone meeting logs for the first time:

A logarithm is just an exponent. The statement \log_b(x) = y is exactly the same fact as b^{\,y} = x — not a new, scary object. Any time you feel lost, rewrite the log in index form and the fog lifts. Stuck on \log_4(64)? Ask "4 to what power is 64?" — that's 3.
You cannot take the log of zero or a negative number. A positive base raised to any power stays positive — 2^{100} is huge, 2^{-100} is a tiny positive sliver, and you can never reach 0 or a negative. So \log_2(-8) and \log_2(0) simply do not exist. Writing one down is a classic invalid step in an exam.

Because in 1614, before calculators, multiplying two big numbers by hand was a slow, error-prone nightmare — and astronomers and navigators had to do it constantly. The Scottish laird John Napier spotted a magic trick hiding in logarithms: since a product of numbers matches a sum of their logs, you can multiply just by adding. Look up two logs in a table, add them, look the answer back up — done. A day's grinding calculation became minutes.

This one idea powered centuries of astronomy, navigation, and engineering. It gave us the slide rule — a ruler with logarithmic markings that multiplies by sliding — the "pocket calculator" that engineers used right up until the 1970s and that helped put astronauts on the Moon. And it is why we still measure earthquakes (the Richter scale), loudness (decibels), and acidity (pH) on logarithmic scales, where every single step means a \times 10 jump. A magnitude-7 quake isn't a bit worse than a magnitude-6 — it's ten times the shaking.