Example 1
- Number: 8
- Base: 2
3
Because 2³ = 8.
Logarithm calculator
Type the base and the number. A logarithm asks which power of the base gives that number. log₂ 8 = 3, because 2³ = 8. log₁₀ 1000 = 3, because 10³ = 1000.
A logarithm undoes exponentiation. You are not asking “what is 10 squared?”, but “to which power should I raise 10 to get 100?”. Answer: 2. The base must be positive and not 1, and the number must be positive too.
Result
Enter a number and a base to see the result.
What it means
Take 100 and base 10. You ask: 10 to which power is 100? 10² = 100, so the logarithm is 2. log₂ 8 = 3, because 2³ = 8. log₁₀ 1000 = 3, because 10³ = 1000. School notation: log₁₀ 100 = 2.
A logarithm undoes exponentiation. On the power page you type 2 and 3 and get 8. Here you know 8 and 2 and want 3. Base 10 is the common logarithm, base e the natural one. Here you type any legal base.
The fields are Number and Base. Number must be greater than zero. Base must be greater than zero and not 1: not 1, not 0, not negative. Type 8 and 2, click Calculate. A comma and a period both parse. If the result is a fraction, you need a non-integer power, and that is fine.
eˣ lives on the exponential page, not here. log_e of e is 1, but bare eˣ has its own Exponent field. A square root is the 1/2 power: √81 = 9, and log₉ 81 = 2.
Without a positive number or a legal base, nothing computes. 8 and 2 still give 3 no matter the header.
log₂ 16 = 4, log₁₀ 100 = 2, log₅ 25 = 2, log₃ 27 = 3. Check 8 with base 2 first.
Where it helps
- School work on powers and logs.
- Checking “which power gives this number?”.
- Thinking about orders of magnitude in a simple school way.
- Moving between multiplying powers and adding exponents.
How to use
- Type the number you want the log of, greater than zero, for example 8.
- Type a base greater than zero and not 1, for example 2.
- Click Calculate. log₂ 8 = 3. log₁₀ 1000 = 3.
- Base 1, 0, or a negative base does not run. Zero or a negative number does not run either.
- Bare 2³ = 8 lives on power. eˣ lives on the exponential page.
log₂ 8 = 3 in this calculator
A logarithm asks which power of the base. log₂ 8 = 3, because 2³ = 8. log₁₀ 1000 = 3, because 10³ = 1000. The number must be positive.
- log
- The calculator result: the exponent. log₂ 8 = 3. log₁₀ 1000 = 3. log₂ 0.25 = −2, because 2⁻² = 0.25.
- e
- Natural-log base, about 2.718. You type it yourself. eˣ lives on the exponential page.
Examples
Example 2
- Number: 1000
- Base: 10
3
Because 10³ = 1000.
Example 3
- Number: 0.25
- Base: 2
−2
Because 2⁻² = 0.25. A negative result means a negative power.
Related calculators
FAQ
What is log₂ 8?
The result is 3, because 2³ = 8. Same example on the page: base 2, number 8. That is not 2 × 8.
What is log₁₀ 1000?
The result is 3, because 10³ = 1000. log₁₀ 100 = 2, because 10² = 100. Common log counts the zeros after the 1.
What is a logarithm?
The inverse of exponentiation. You ask which power of the base gives the number. log₂ 8 looks for the 3 in 2³.
Can the base be 1?
No. 1 to any power stays 1, so the logarithm has no meaning. 0 and a negative base fail too.
Can the number be negative?
Not in the reals. The number must be greater than zero. A minus in the number field is rejected.
What are log₅ 25 and log₃ 27?
log₅ 25 = 2, because 5² = 25. log₃ 27 = 3, because 3³ = 27. log₂ 16 = 4, because 2⁴ = 16.
How is this different from a power?
A power from 2 and 3 gives 8. The logarithm of 8 with base 2 gives 3. The same three numbers, the inverse path.
Where is the natural log?
Type base e, about 2.718. eˣ itself lives on the exponential page. Here the unknown is the exponent.
Can the result be a fraction?
Yes. log₂ 10 is not an integer. The result is a decimal, it does not hunt for a common fraction.
Is √81 the same computation?
√81 = 9 lives on the square-root page. log₉ 81 = 2, because 9² = 81. Another question, another calculator, the same 81.
Knowledge sources
The calculator computes the same formula as the definition below.
Page updated in 2026.