Case 1
- Value x
- 1000
- Base b
- 10
3.000000
Open with these valuesLogarithm Calculator
3.000000
Result: 3.000000A logarithm answers one question: to what power must the base be raised to reach your value? Because 10³ = 1000, the logarithm of 1000 to base 10 is 3. Any base works — the calculator divides ln(value) by ln(base), which is the change-of-base rule.
The numbers at a glance
Held fixed: Value x 1,000.000000.
| Base b | Result |
|---|---|
| 2.500000 | 7.538825 |
| 5.000000 | 4.292030 |
| 7.500000 | 3.428331 |
| 10.000000Your value | 3.000000 |
| 12.500000 | 2.734955 |
| 15.000000 | 2.550822 |
| 17.500000 | 2.413442 |
| 20.000000 | 2.305865 |
Worked examples
Case 2
- Value x
- 8
- Base b
- 2
3.000000
Open with these valuesCase 3
- Value x
- 50
- Base b
- 5
2.430677
Open with these valuesHow it's calculated
log_b(x) = ln(x) ÷ ln(b)
- StepEnter the value you want the logarithm of; it has to be positive.
- StepEnter the base — 10 and 2 are the usual ones, e ≈ 2.71828 gives ln.
- ResultRead the exponent that turns the base into your value.
What this number means
The inverse of raising to a power
log_b(x) asks which exponent turns the base into your value. Because 10³ = 1000, the logarithm of 1000 to base 10 is 3, and because 2⁵ = 32 the logarithm of 32 to base 2 is 5.
The base is a field, not a fixed setting
Base 10 is the common log behind pH, decibels and the Richter scale, base 2 belongs to computing, and e ≈ 2.71828 gives the natural log ln. Any other positive base runs through the change-of-base rule ln(x) ÷ ln(b).
Base 1 is not defined
The change-of-base rule divides by ln(base), and ln(1) is zero. Base 1 could never produce anything but 1 either, so the calculator leaves the result blank there.
Zero and negative values have no logarithm
No real power of a positive base ever gives zero or a negative number. The curve drops without limit as the value approaches zero, so enter any value above zero, however small.
Commonly misread
log₁₀(1000) and ln(1000) are the same number.
They are the same value read to two different bases: log₁₀(1000) is 3, while ln(1000) is 6.907755.
A number below 1 has no logarithm.
It does: log₂(0.5) is −1, because 2 raised to −1 is 0.5. Only zero and negative values are outside the real logarithm.
Base 1 simply returns the value itself.
Base 1 is not defined, because the change-of-base rule would divide by ln(1) = 0. The calculator shows no result.
An awkward base such as 5 needs its own function.
The change-of-base rule covers it: 3.912023 ÷ 1.609438 puts 50 to base 5 at 2.430677.
Reference table
| Value, base | Natural log ln(x) | Logarithm |
|---|---|---|
| 0.5, 2 | -0.693147 | -1 |
| 7, 7 | 1.945910 | 1 |
| 8, 2 | 2.079442 | 3 |
| 27, 3 | 3.295837 | 3 |
| 50, 5 | 3.912023 | 2.430677 |
| 100, 10 | 4.605170 | 2 |
| 1000, 10 | 6.907755 | 3 |
Questions
What is a logarithm?
A logarithm is the exponent that turns a base into a given number. log_b(x) asks: to what power must b be raised to get x? It is the inverse of raising to a power, so because 2⁵ = 32, the logarithm of 32 to base 2 is 5.
How do I calculate a logarithm to any base?
Use the change-of-base rule: log_b(x) = ln(x) ÷ ln(b). Take the natural logarithm of the value, take the natural logarithm of the base, and divide. For base 10 and 1000 that is 6.907755 ÷ 2.302585 = 3, because 10³ = 1000.
What is the natural logarithm (ln)?
The natural logarithm is the logarithm to base e ≈ 2.71828, written ln(x). Enter e as the base and this calculator returns exactly that; the middle column of the table above shows ln for each row. Natural logs run through growth, decay and calculus because e has uniquely simple rate-of-change behaviour.
Why can't the base be 1?
The change-of-base rule divides by ln(base), and ln(1) is zero, so base 1 would mean dividing by zero. Base 1 could never produce anything but 1 either, so there is no exponent to find. The calculator leaves the result blank there.
Why must the value be positive?
No real power of a positive base ever gives zero or a negative number, so the logarithm of such a value does not exist among the real numbers. The curve drops without limit as the value approaches zero. Enter any value above zero, however small.
What are the most common bases?
Base 10 is the common log, behind pH, decibels and the Richter scale. Base 2 belongs to computing and information theory, and base e ≈ 2.71828 is the natural log. This calculator handles all three, and any other positive base, with the same formula.
Sources and last check
Information, not professional advice.