Common log: log₁₀(1000)
- Input
- Number = 1000, Base = 10
- Working
- Ask: 10 to what power equals 1000? 10³ = 10 × 10 × 10 = 1000. Change-of-base check: ln(1000) / ln(10) = 6.9078 / 2.3026 = 3.
- Result
- log₁₀(1000) = 3
Logarithm Calculator
Find logarithms in any base — common log, natural log, or your own — using the change-of-base rule, with the power check shown beside every answer. Runs entirely in your browser — your numbers never leave this device.
Quick base
Logarithm
3
log base 10 of 1,000 = 3
- Natural log ln(x)
- 6.907755279
- Common log log₁₀(x)
- 3
- Check: 10 ^ result
- 1000
How it works
Definition: log_b(x) = y means b^y = x | Change of base: log_b(x) = ln(x) / ln(b) = log₁₀(x) / log₁₀(b) | Rules: log(a×c) = log a + log c, log(a/c) = log a − log c, log(aⁿ) = n × log a
| x | log₁₀(x) | Reading |
|---|---|---|
| 1 | 0 | 10⁰ = 1 |
| 10 | 1 | 10¹ = 10 |
| 100 | 2 | 10² = 100 |
| 1,000 | 3 | 10³ = 1,000 |
| 10,000 | 4 | 10⁴ = 10,000 |
| 100,000 | 5 | 10⁵ = 100,000 |
| 1,000,000 | 6 | 10⁶ = 1,000,000 |
A logarithm answers the question 'to what power must I raise the base to get this number?' log₂(8) = 3 because 2³ = 8. Enter the number and the base and the calculator evaluates the log using the change-of-base formula — dividing the natural log of the number by the natural log of the base — which works for any valid base, then shows the power check (base raised to the answer) so the result verifies itself.
Three bases dominate in practice. Base 10 (written log) is used for scales like pH, decibels and the Richter scale; base e (written ln, the natural log) appears throughout calculus, growth and decay; and base 2 appears in computing, where log₂ tells you how many doublings — or bits — a value represents. Shortcuts for all three are provided, but any other positive base works too.
The domain limits are strict and the calculator enforces them openly: the number must be greater than zero (the log of 0 or of a negative number has no real answer), the base must be greater than zero, and the base cannot equal 1, because 1 raised to any power is always 1. Results are given to four decimal places, with exact whole-number answers like log₁₀(1000) = 3 shown without trailing decimals.
Worked examples
Log base 2: log₂(8)
- Input
- Number = 8, Base = 2
- Working
- Ask: 2 to what power equals 8? 2¹ = 2, 2² = 4, 2³ = 8. So the power is 3. Change-of-base check: ln(8) / ln(2) = 2.0794 / 0.6931 = 3.
- Result
- log₂(8) = 3
Log base 3: log₃(81)
- Input
- Number = 81, Base = 3
- Working
- Powers of 3: 3¹ = 3, 3² = 9, 3³ = 27, 3⁴ = 81. The matching power is 4, so the log is 4.
- Result
- log₃(81) = 4
Frequently asked questions
What is the difference between log and ln?
Both are logarithms; they differ only in base. 'log' without a base shown means base 10 (the common logarithm), while 'ln' means base e ≈ 2.71828 (the natural logarithm). So log(1000) = 3 but ln(1000) ≈ 6.908. Always check which one a formula expects — mixing them up is the most common log error.
Why is there no logarithm of zero or a negative number?
Because no power produces them. A positive base raised to any real power is always positive — it can get very close to zero but never reach it, and it can never go negative. So log(0) and log(−5) are undefined in real numbers, and the calculator reports that instead of inventing a value.
What is the change-of-base formula used for?
Most calculators and software only compute base-10 and natural logs. The change-of-base formula, log_b(x) = ln(x) / ln(b), converts any base into those two, which is how this calculator handles bases like 2, 3 or 7. It is also a standard exam technique for evaluating logs by hand.
Why can't the base be 1?
Because 1 raised to any power is always 1. That means log₁(1) could be anything and log₁ of any other number could be nothing — there is no single answer, so base 1 is excluded by definition. The same reasoning excludes base 0 and negative bases.
What does log₂ tell me in computing?
log₂(x) tells you how many times x can be halved before reaching 1, or equivalently how many bits are needed to represent x values. log₂(8) = 3 means 3 bits cover 8 distinct values (000 to 111), and log₂(1024) = 10 means 10 doublings get you from 1 to 1024.
Does this log calculator send my inputs anywhere?
No. It runs 100% in your browser and nothing is uploaded — the numbers and bases you enter are processed locally on your device and forgotten when you close the page.
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