Evaluate a positive argument at any valid logarithm base. The calculation trail makes the reported logarithm easier to reproduce.
A logarithm answers an exponent question: log_b(x) is the power y satisfying bʸ = x. Bases must be positive and unequal to 1, while real-valued arguments must be positive.
For this page, logarithm is interpreted under the stated convention and input order.
Logarithms invert exponential growth, compress wide numerical ranges, solve for time in compound processes, and appear throughout science, computing, and statistics.
log(a + b) does not split into log a + log b; that rule belongs to products. Keep track of the base, because log₂(8) and log₁₀(8) are different values. If the Logarithm assumptions do not fit, consider inverse exponent operation.
Compare the result with the worked example's scale before relying on the reported logarithm.
log₁₀(1000) = 3 because 10³ = 1000. For a less obvious base, the change-of-base formula ln(x)/ln(b) supplies the exponent. This Logarithm example can be compared with powers of ten.
For recognizable powers, ask what exponent on the base produces the argument. Otherwise divide natural logarithm of the argument by natural logarithm of the base.
Source values may arrive in a different order from the form. Map them explicitly to argument x and base, normalize units, and retain enough precision for the next step after logarithm.
For bases above one, increasing the argument increases the logarithm; bases between zero and one reverse that direction. Values between zero and one have negative logs when the base exceeds one. Watching this response separates a data-entry mistake from an unexpected but valid value.
If argument x and base are exact counts, more result digits may be meaningful than when they are measured approximations. Let the least certain source guide the final presentation.
Logarithms undo exponentiation. Roots also undo powers in a restricted sense, but a logarithm solves for the exponent rather than for the powered base value. This page deliberately reports only the former interpretation.
Label the output as logarithm in notes or tables. Store argument x and base beside it when the result will be reused in a later stage.
No finite real exponent on a positive base produces zero.
It is the logarithm with base e, written ln.
Yes. The logarithm then decreases as the argument increases.
Zero for every valid base because b⁰ = 1.
Every power of 1 is 1, so it cannot represent other positive arguments uniquely.
No; complex logarithms can, but this page evaluates real logarithms.