Calculate the logarithm of a number in any base, along with its natural logarithm (ln), common logarithm (log₁₀) and binary logarithm (log₂).
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With number (x) 1,000, base (b) 10, the logarithm is 3.
| Number (x) | 1,000 |
|---|---|
| Base (b) | 10 |
| Logarithm | 3 |
|---|---|
| Natural logarithm (ln) | 6.907755279 |
| Common logarithm (log₁₀) | 3 |
| Binary logarithm (log₂) | 9.965784285 |
A logarithm answers the question “what power of the base gives this number?” For example log₁₀(1000) = 3 because 10³ = 1000. Enter the number and the base. Any positive base except 1 works; common choices are 10, 2 and e (about 2.718281828). The number must be positive, because a positive base raised to any real power is never zero or negative.
log_b(x) = ln(x) ÷ ln(b). The logarithm is the power you raise b to in order to get x: b^y = x. A logarithm answers the question “what power of the base gives this number?” For example log₁₀(1000) = 3 because 10³ = 1000. Enter the number and the base. Any positive base except 1 works; common choices are 10, 2 and e (about 2.718281828). The number must be positive, because a positive base raised to any real power is never zero or negative.
With number (x) 1,000, base (b) 10, the logarithm is 3. Natural logarithm (ln): 6.907755279; Common logarithm (log₁₀): 3; Binary logarithm (log₂): 9.965784285.
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log_b(x) = log_k(x) ÷ log_k(b) for any convenient base k. Calculators usually use k = e, so log₂(32) = ln 32 ÷ ln 2 = 5.
Raising a positive base to any real power always gives a positive result, so no real exponent produces zero or a negative number. As x approaches zero the logarithm falls towards minus infinity.
Raise the base to the result. If log₁₀(x) = 2.5, then x = 10^2.5 ≈ 316.2. The exponent calculator does this step.