The formula
How it works
A logarithm answers the question “what power do I raise this base to, to get my number?” It is the inverse of an exponent. Enter a number and a base to find its log, plus the common (base 10) and natural (base e) logs alongside.
FAQ
What are common and natural logs?
The common log uses base 10 and is written log; the natural log uses the base e ≈ 2.718 and is written ln. Both are just logarithms with a particular, widely used base.
Why can’t I take the log of zero or a negative?
No power of a positive base ever gives zero or a negative number, so the logarithm is only defined for numbers greater than zero. The calculator flags anything else as undefined.
Why can’t the base be 1?
Every power of 1 is just 1, so raising 1 to any exponent can never produce any other number — there is no unique answer, which is why a base of 1 is excluded.
What does a negative logarithm mean?
A negative result means the number is between 0 and 1, since raising a base greater than 1 to a negative power gives a fraction — for example log base 10 of 0.01 is −2.
How is the change-of-base formula used here?
The calculator computes any log by dividing the natural log of the number by the natural log of the base, \(\log_b(x) = \ln x / \ln b\), which is how any base can be handled with a single method.
What is the log of a number to its own base?
It is always 1, because any base raised to the power 1 simply gives that base back — for example log base 7 of 7 equals 1.
What is the log of 1, whatever the base?
It is always 0, since any positive base raised to the power 0 equals 1 — so log 1 is 0 no matter which base you pick.
About the log calculator
This calculator finds the logarithm of a number to any base, and also shows the common log (base 10) and natural log (base e). A logarithm is the mirror image of a power: where an exponent asks “what do I get if I raise this base to that power?”, a logarithm asks “what power gets me to this number?”. That reverse view makes logs the natural tool for anything that grows or shrinks by multiplying.
How to use it
Enter the number you want the logarithm of, then the base. The calculator returns the log to that base, along with the base-10 and natural logs for comparison. For example, the log of 1000 to base 10 is 3, because 10 raised to the power 3 is 1000. Likewise the log of 8 to base 2 is 3, since 2 × 2 × 2 = 8. The number must be positive; the base must be positive and not equal to 1.
The formula
A logarithm is defined by : the log is the exponent that turns the base into the number . To compute a log to any base, the calculator uses the change-of-base rule, , dividing two natural logs. The main log laws follow from the exponent rules: turns multiplication into addition, which is what made log tables so powerful.
Where it is used
Logarithms tame numbers that span a huge range. The Richter scale for earthquakes, the decibel scale for sound and the pH scale for acidity are all logarithmic, so each step is a tenfold change. Scientists use natural logs in models of growth and decay, and computer scientists use base-2 logs to measure how algorithms scale. Before calculators, logarithms were the everyday trick for multiplying large numbers by adding instead.