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- Finds the logarithm of any positive number in base 10, e, 2 or any custom base.
- Shows ln, log₁₀ and log₂ at the same time.
- Uses the change of base formula and shows each step.
- Also works backwards as an antilog calculator (bʸ).
How to use the log calculator
Choose log or antilog
Log finds the exponent. Antilog raises the base to a power.
Enter the number
Any number greater than 0, such as 100 or 0.001.
Pick the base
10 for common log, e for natural log (ln), 2 for binary log, or any other positive base except 1.
Read the answer
You get the result, its exponential form and the change of base working.
How logarithms work
Definition
logᵦ(x) = y means bʸ = x
A logarithm answers the question: what power do I raise the base to, to get x?
Change of base and key rules
logᵦ(x) = ln(x) ÷ ln(b)
log(a × c) = log a + log c
log(a ÷ c) = log a − log c
log(aⁿ) = n × log a
These rules let you simplify products, quotients and powers before calculating.
Example 1: common logs
Base 10.
- log(100) 2
- log(50) 1.69897
- log(0.001) −3
Numbers between 0 and 1 have negative logarithms.
Example 2: other bases
Using change of base.
- ln(10) 2.302585
- log₂(1024) 10
- log₃(81) 4
For log₃(81): ln 81 ÷ ln 3 = 4, because 3⁴ = 81.
Example 3: antilog
Raise the base to the exponent.
- 10² 100
- e¹ 2.718282
An antilog undoes a logarithm with the same base.
Where a log calculator helps
Algebra and calculus
Solve exponential equations and check log rules.
Chemistry (pH)
pH is the negative log of the hydrogen ion concentration.
Sound and earthquakes
Decibels and the Richter scale use base-10 logarithms.
Computer science
Binary logs measure bits and the speed of algorithms.
Raise numbers to any power with the exponent calculator, handle very large or small results with the scientific notation calculator, and model growth over time with the compound interest calculator.
Log calculator FAQ
What is a logarithm?
A logarithm is the exponent you need. log₁₀(1000) = 3 because 10³ = 1000.
What is the difference between log and ln?
log usually means base 10, and ln means the natural log with base e (about 2.71828).
Why can I not take the log of 0 or a negative number?
No power of a positive base gives 0 or a negative number, so the logarithm is undefined.
How do I calculate a log with a different base?
Use the change of base formula: logᵦ(x) = ln(x) ÷ ln(b). The calculator does this for you.
What is an antilog?
It reverses a logarithm. The antilog of 2 in base 10 is 10² = 100.
What are the characteristic and mantissa?
For a base-10 log, the characteristic is the whole-number part and the mantissa is the decimal part. They were used with printed log tables.