1
Enter the value
Use a positive whole number, decimal, or scientific-notation value.
Free Logarithm Calculator
A logarithm calculator finds the exponent that raises a selected base to a positive value. Calculate common, binary, natural, or custom-base logarithms and see the change-of-base formula with each substituted step.
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Change-of-base calculator
Enter a value and base
Result
Result: 3
log_10(1000) = 3
Change of base
ln(1000) / ln(10) = 3
Step-by-step explanation
- 1Use the change-of-base formula: log_b(x) = ln(x) / ln(b).
- 2Substitute x = 1000 and b = 10.
- 3Divide ln(1000) by ln(10) to get 3.
- 4Check: 10 raised to 3 equals 1000 (within rounding).
How to Calculate a Logarithm
2
Choose the base
Select 10, 2, e, or enter any positive custom base except 1.
3
Check the exponent
Review the change-of-base work and confirm the base raised to the result.
Common Logarithm Rules
Product rule
log_b(MN) = log_b(M) + log_b(N)
Quotient rule
log_b(M/N) = log_b(M) - log_b(N)
Power rule
log_b(M^p) = p · log_b(M)
Inverse rule
log_b(b^x) = x and b^(log_b x) = x
Logarithm Calculator FAQ
What is a logarithm calculator?
A logarithm calculator finds the exponent needed to raise a chosen base to a positive value. For example, log base 10 of 1000 is 3 because 10 cubed is 1000.
How do you calculate a logarithm with a different base?
Use the change-of-base formula: log base b of x equals ln(x) divided by ln(b). The value x must be positive, while b must be positive and not equal to 1.
What is the difference between log and ln?
Log commonly means base 10, while ln means the natural logarithm with base e, approximately 2.71828. Both answer the same exponent question using different bases.
Can a logarithm be negative?
Yes. A logarithm is negative when the positive input lies between 0 and 1 for a base greater than 1. For example, log base 10 of 0.01 is -2.
Why can the input not be zero or negative?
Real-number logarithms are defined only for positive inputs. Zero and negative values do not have real logarithms, so this calculator flags them instead of returning a misleading result.