Formula
log_b(x) = ln(x) ÷ ln(b)antilog: x = b^y
- x > 0
- b > 0 and b ≠ 1
- ln is the natural logarithm (base e ≈ 2.71828)
A logarithm answers: to what power must the base be raised to get x?
Worked examples
log₁₀ of 1000
- What do you want?
- Logarithm: log base b of x
- Number x
- 1,000
- Base b
- 10
log base 10 of 1,000
3
Natural log, ln(x): 6.907755279
Common log, log₁₀(x): 3
Binary log, log₂(x): 9.9657842847
log₂ of 256
- What do you want?
- Logarithm: log base b of x
- Number x
- 256
- Base b
- 2
log base 2 of 256
8
Natural log, ln(x): 5.5451774445
Common log, log₁₀(x): 2.4082399653
Binary log, log₂(x): 8
How it works
log base b of x is the exponent y for which b^y = x. Because 10³ = 1000, log₁₀(1000) = 3, and because 2⁸ = 256, log₂(256) = 8. The calculator uses the change-of-base rule log_b(x) = ln(x) ÷ ln(b), so it works for any positive base other than 1.
Logarithms are only defined for positive numbers, and the base cannot be 1 because every power of 1 is 1. The natural logarithm ln uses the constant e, the common logarithm log₁₀ is used for scales such as pH and decibels, and log₂ appears throughout computing. The antilogarithm reverses a log: the antilog of 3 in base 10 is 10³ = 1000.
To work with powers directly use the Exponent Calculator, and for roots the Square Root Calculator. Longer expressions that mix logs with other operations can be typed into the Scientific Calculator.
Assumptions and limits
- Real logarithms only: x must be positive and the base positive and not 1.
- Results are accurate to about 12 significant digits.