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A logarithm answers the question "what power do I raise the base to, to get this number?" log₁₀(1000) is 3 because 10³ = 1000. It's the inverse of exponentiation and the backbone of scales like pH, decibels and the Richter scale.
Enter a number and a base to get its logarithm, along with the natural log, log₁₀ and log₂.
Logarithm Calculator: step-by-step worked example
10 to what power gives 500? That is log base 10 of 500.
- Number (x)
- 500
- Base (b)
- 10
- 1
Natural log of the number
ln x
ln 500 = 6.2146
- 2
Natural log of the base
ln b
ln 10 = 2.3026
- 3
Change-of-base rule
log_b(x) = ln x ÷ ln b
6.214608 ÷ 2.302585 = 2.699
- 4
Check: raise the base
b^(log_b x) = x
10^2.69897 = 500
- log₂8.9658
- ln6.2146
- log₁₀2.699
Good to know: A log answers “how many times do I multiply the base?”: log₁₀ 500 ≈ 2.7 means 500 lies between 10² and 10³. On calculators “log” is usually base 10 and “ln” base e. log(a × b) = log a + log b.
Logarithm Calculator — how it works
What a logarithm is
log_b(x) is the exponent that turns b into x: if b^y = x, then log_b(x) = y. So log₂(8) = 3 because 2³ = 8.
Change of base
Calculators compute any base from natural logs using the change-of-base rule:
log_b(x) = ln(x) ÷ ln(b)
That's how this tool handles an arbitrary base — and why it can show ln, log₁₀ and log₂ side by side for the same number.
The common bases
| Notation | Base | Used for |
|---|---|---|
| ln | e ≈ 2.718 | calculus, growth/decay |
| log or log₁₀ | 10 | pH, decibels, scientific notation |
| log₂ | 2 | computer science, information theory |
The rules that make logs useful
Logarithms turn multiplication into addition: log(x·y) = log(x) + log(y), log(x/y) = log(x) − log(y), and log(xⁿ) = n·log(x). These identities are why log scales compress huge ranges and why they were once used for hand multiplication.
Where it helps
Solving exponential equations, pH and decibel problems, half-life and compound-growth math, and algorithm analysis. The input must be positive — the log of zero or a negative number is undefined.
Worked example
Take log₁₀(1000): you need the power of 10 that gives 1000, which is 3. For a non-round base, say log₂(50), the change-of-base rule gives ln(50) ÷ ln(2) ≈ 3.912 ÷ 0.693 ≈ 5.644 — meaning 2^5.644 ≈ 50. The tool also reports ln(50) ≈ 3.912 and log₁₀(50) ≈ 1.699.
Logarithm table: log₁₀, ln and log₂
| x | log₁₀ x | ln x | log₂ x |
|---|---|---|---|
| 1 | 0 | 0 | 0 |
| 2 | 0.301 | 0.6931 | 1 |
| 3 | 0.4771 | 1.0986 | 1.585 |
| 4 | 0.6021 | 1.3863 | 2 |
| 5 | 0.699 | 1.6094 | 2.3219 |
| 6 | 0.7782 | 1.7918 | 2.585 |
| 7 | 0.8451 | 1.9459 | 2.8074 |
| 8 | 0.9031 | 2.0794 | 3 |
| 9 | 0.9542 | 2.1972 | 3.1699 |
| 10 | 1 | 2.3026 | 3.3219 |
| 20 | 1.301 | 2.9957 | 4.3219 |
| 50 | 1.699 | 3.912 | 5.6439 |
| 100 | 2 | 4.6052 | 6.6439 |
| 1,000 | 3 | 6.9078 | 9.9658 |
Change of base: log_b x = ln x ÷ ln b.
Frequently asked questions
What is a logarithm?+
The logarithm log_b(x) is the exponent to which the base b must be raised to produce x. Since 10³ = 1000, log₁₀(1000) = 3.
What is the difference between log and ln?+
"log" usually means base 10, while "ln" is the natural log with base e ≈ 2.718. This tool shows both for any number you enter.
How does change of base work?+
log_b(x) = ln(x) ÷ ln(b). This lets a calculator find a logarithm in any base using natural logs, which is exactly how this tool computes it.
Why can’t I take the log of a negative number?+
No real power of a positive base gives a negative result, so the logarithm of zero or a negative number is undefined in the real numbers. The input must be positive.
What is log₂ used for?+
Log base 2 counts how many times a number can be halved, so it appears in computer science, information theory (bits) and the analysis of algorithms.
About this page
- Publisher:
- Kalkoro — Teknolojig Bilişim Teknolojileri A.Ş., İstanbul
- Written by:
- Ahmet Tur — Editor (Math, date & time, unit conversion, physics & chemistry, everyday tools)
- Reviewed by:
- Kübra Yıldız — Editor
Parameters come from official sources; the last update date is shown at the bottom of the page.
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