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.
Review Argument x inside the Logarithm relation. Hold Base steady while testing Base. A rough Logarithm gives Logarithm an independent scale check. Keep the revised Logarithm inputs beside their own Logarithm.
Logarithms invert exponential growth, compress wide numerical ranges, solve for time in compound processes, and appear throughout science, computing, and statistics.
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.
Link Argument x to its Logarithm role. Link Base to its Logarithm role. The retained Logarithm formula identifies the Logarithm model.
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.
Substitute the Logarithm solution into Argument x. This Logarithm check rejects false Logarithm branches and forbidden denominators.
Choose an easy Argument x value before running Logarithm. Predict Base, then compare it with the Logarithm output.
Units and conventions belong with a Logarithm answer. Confirm that Argument x and Base use the intended interpretation, then label Logarithm the same way. A numerically correct Logarithm can still answer the wrong Logarithm question when that context changes.
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.