What Is a Logarithm? A Plain-English Guide With Examples

Exponents and logarithms are two sides of the same coin. When you write 10 to the power 3 = 1000, you start with the exponent and find the result. A logarithm flips that around: you know the result (1000) and the base (10), and you want the exponent. That hidden exponent is the logarithm. Once you see logs as "the exponent detective," most of the mystery disappears.

What does a logarithm actually mean?

A logarithm is the inverse of exponentiation. The statement log base b of x = y is exactly the same fact as b to the power y = x, just rearranged. The base is the number being multiplied by itself, x is the number you are taking the log of (the argument), and y is the answer, which is the exponent.

Read it out loud as a question and it clicks: "log base 2 of 8" is asking "2 raised to what power gives 8?" Since 2 times 2 times 2 equals 8, the answer is 3. So log base 2 of 8 = 3. The same logic covers the headline example: log base 10 of 1000 = 3 because you multiply 10 by itself three times to reach 1000.

Notice the last two rows. The log of 1 is always 0 in any base, because raising a base to the power 0 gives 1. And the log of the base itself is always 1, because a base to the power 1 is just the base. These two facts are worth memorising as sanity checks.

What are common logs and natural logs?

Two bases show up so often they get their own shorthand. The common logarithm uses base 10 and is usually written simply as log with no base shown. The natural logarithm uses base e (Euler's number, approximately 2.71828) and is written ln. So ln(x) just means log base e of x.

• Common log (log, base 10): the default on most calculators and in science, engineering, and the decibel and pH scales.
• Natural log (ln, base e): the natural choice in calculus, continuous growth, and finance, because e is the constant that describes smooth compounding.
• Binary log (base 2): common in computer science, since data and complexity often double or halve.

Why base e? It appears naturally whenever something grows or decays continuously, which is why it underpins topics like exponential growth and compound interest. You do not need to fully grasp e to use ln, but it helps to know it is a specific irrational number, not a made-up label.

What are the logarithm rules?

A handful of rules let you break complicated logs into simple pieces. They all come straight from the laws of exponents, which is why they work in every base. Here are the ones you will actually use.

The product rule turns multiplication into addition, the quotient rule turns division into subtraction, and the power rule pulls an exponent out to the front as a multiplier. That last one is the secret behind why logs were historically used to do hard multiplication by hand, and why they still power log scales today.

How does the change of base formula work?

Most calculators only have buttons for log (base 10) and ln (base e). The change of base formula lets you compute a log in any base using those two. The formula is log base b of x = ln(x) / ln(b). You can use log instead of ln on top and bottom and get the same answer, as long as both use the same base.

Worked example: find log base 2 of 50 without a base-2 button.

1. Write the change of base setup: log base 2 of 50 = ln(50) / ln(2).
2. Compute ln(50), which is about 3.912.
3. Compute ln(2), which is about 0.693.
4. Divide: 3.912 / 0.693, which is about 5.644.
5. Sanity check: 2 to the power 5 is 32 and 2 to the power 6 is 64, so an answer near 5.64 sits sensibly between them.

Where are logarithms used in real life?

Logarithms are not just a classroom exercise. They appear anywhere numbers span a huge range and a straight scale would be useless. Compressing that range into something readable is exactly what a log scale does.

• Earthquakes: the Richter scale is logarithmic, so a magnitude 6 releases roughly 32 times more energy than a magnitude 5.
• Sound: decibels measure loudness on a base-10 log scale, which is why a small dB increase feels large.
• Chemistry: pH is the negative log of hydrogen ion concentration.
• Finance and growth: returns and continuous compounding lean on natural logs, closely tied to exponential growth.
• Computing: algorithm speed is often described with base-2 logs, because splitting a problem in half repeatedly is a logarithmic process.

Common mistakes to avoid

Logs are simple once the relationship clicks, but a few errors trip almost everyone at first. Watch for these.

• Splitting a log of a sum. log(a + b) is NOT log a + log b. The product rule only applies to multiplication inside the log, not addition.
• Taking the log of zero or a negative number. These are undefined for real logs, because no real exponent turns a positive base into zero or a negative result.
• Confusing log and ln. On many calculators log means base 10 and ln means base e. Using the wrong one gives a wrong answer, so check which base a problem expects.
• Forgetting the base is fixed above 0 and not 1. A valid logarithm base must be positive and not equal to 1, otherwise the exponent relationship breaks down.
• Misplacing the power rule. log(a^n) = n log a moves the exponent out front. It does not become (log a) to the power n.

If you ever doubt an answer, convert it back to exponent form. For log base 10 of 1000 = 3, ask whether 10 to the power 3 really equals 1000. It does, so the log is correct. This reverse check catches most slips. For the reverse operation, turning an exponent back into a number, you can use an antilog calculator.

Good to know: logs and their neighbours

Logarithms sit next to several related ideas. Understanding how they connect makes the whole family easier. Logs undo exponents, exponents describe repeated multiplication, and both show up throughout sequences and growth. If you are brushing up on the surrounding math, the closely related geometric sequence formula uses repeated multiplication in a way that logs can neatly unwind, and scientific notation leans on base-10 powers, the same base as the common log.

You can verify the official value and history of Euler's number through a primary reference such as the NIST Digital Library of Mathematical Functions. Sticking to authoritative sources like that keeps your understanding grounded rather than guesswork.

A logarithm is nothing more than an exponent asked in reverse: given a base and a result, it tells you the power that connects them. Master the three log rules, the change of base formula, and the difference between log and ln, and you can handle almost any logarithm you meet, from a homework problem to a decibel reading. Keep the calculator above handy, use the exponent check to verify your work, and logs will feel routine in no time.