A logarithm answers the question “what power of the base gives me this number?” For example, log(1000) = 3 because 10³ = 1000. Get that one idea and every logarithm you meet is just the exponent asked for in a different outfit.

Key Takeaways

A logarithm is the inverse of an exponent: log(1000) = 3 because 10³ = 1000.

The base-10 log (written log) counts powers of ten and is how scientists compress scales.

The natural log (written ln) is base e ≈ 2.71828, the number behind growth.

log(1) = 0 and log(10) = 1 for any well-formed base, which is a fast sanity check.

What is a logarithm?

A logarithm (shortened to log) is the inverse operation of exponentiation. Where a power answers “what is 10³?”, a logarithm asks the reverse: “what power do I raise 10 to, to get the number facing me?” So log(1000) is “the power of 10 that equals 1000”, and since 10³ = 1000, the answer is 3.

Worked example 1: reading a base-10 log. log(100) = 2 because 10² = 100. log(10) = 1 because 10¹ = 10. log(1) = 0 because 10⁰ = 1. And log(0.01) = -2 because 10⁻² = 0.01. Type log(100) on the log calculator to confirm it returns 2.

How to read a decimal log

Logs do not always come out to whole numbers, and the decimal tells you how far between two powers of ten a number sits. Because 100 = 10² and 1000 = 10³, a number between them, like 300, has a log between 2 and 3: log(300) ≈ 2.477. The whole part (2) tells you the order of magnitude; the fraction (0.477) pinpoints where it falls within that decade. That is the same reasoning behind scientific notation, where 300 is written 3.0 × 10².

The base-10 log versus the natural log

The base you choose for a log is written as a small subscript, but on everyday calculators two bases dominate and each has a dedicated key:

log is base 10 (common logarithm). This calculator follows standard scientific-calculator convention: pressing log computes base 10, so log(1000) = 3.
ln is base e (natural logarithm), where e ≈ 2.71828.... Press ln to compute base e, so ln(e) = 1.

The two are interchangeable through a change of base: any log can be written in terms of any other. That is why a site with both a log calculator and a natural log calculator can offer “any base” and “base e” side by side without duplicating the maths.

Worked example 2: comparing bases. log(100) = 2 (base 10), but ln(100) = 4.60517018599 (base e). Both answer the same question, “what power of the base?”, just with different bases, and both are quickly reproduced on the log and exponent rules page.

Logs compress huge ranges

The property that makes logs useful is that they turn multiplication into addition: log(a × b) = log(a) + log(b). Instead of juggling enormous numbers, you add their logs. That is why logarithmic scales appear wherever numbers span many powers of ten, from the Richter scale for earthquakes to the decibel scale for sound and the scientific notation used across physics.

Worked example 3: a power rule check. The power rule says log(aⁿ) = n × log(a). Verify it: log(10⁶) = log(1,000,000) = 6, and 6 × log(10) = 6 × 1 = 6. Both methods agree.

Using the change-of-base formula

Calculators only offer two dedicated log keys, log (base 10) and ln (base e), but you can find a log in any base with one formula: logᵦ(x) = log(x) / log(b). Divide the base-10 log of the number by the base-10 log of the base you actually want, and the base-10 parts cancel out to leave the answer in the base you asked for.

Worked example 4: log base 2 of 8. Suppose you need log₂(8), the power of 2 that gives 8. By the formula, log₂(8) = log(8) / log(2) = 0.90309 / 0.30103 = 3. That checks out directly too, since 2³ = 8. The same trick works with ln in place of log, because the base-10 parts cancel either way: ln(8) / ln(2) = 2.07944 / 0.69315 = 3.

This is also how logs help solve exponential equations. To solve 10ˣ = 500 for x, take the log of both sides: x = log(500). Since 500 sits between 100 (log = 2) and 1000 (log = 3), the answer should land between 2 and 3, and it does: log(500) ≈ 2.699. Plug 10^2.699 into the exponent calculator and it lands back near 500, confirming the round trip.

Frequently asked questions

What is a logarithm in simple terms?

It is the exponent asked for backwards. log(x) is “what power of the base gives me x?”, so log(1000) = 3 because 10³ = 1000.

What is the difference between log and ln?

log is base 10; ln is base e ≈ 2.71828. On this calculator the log key always means base 10 and the ln key always means natural.

Why is log(1) always 0?

Because any base raised to the power 0 equals 1: 10⁰ = 1, e⁰ = 1, and so on. The logarithm reads that back as 0.

Can you take the log of a negative number?

Not as a real number, because no real power of a positive base gives a negative result. Entering log(-5) returns an error, and the calculator error messages guide explains why.