Logarithm Calculator
log base b of x = ln(x) / ln(b). The value and base must be positive, and the base cannot be 1.
Choose the value and base
Enter the value and the base. A logarithm answers an exponent question. log base 10 of 100 asks what power of 10 gives 100. The answer is 2.
Use this page for common logs, natural logs, custom bases and exponent checks. If your class writes log without a base, confirm what convention it uses. Calculator buttons often use log for base 10 and ln for base e.
What a logarithm answers
The result is the exponent needed on the base to produce the value. If log base 2 of 8 is 3, that means 2^3 = 8.
The base and value both have restrictions. The value must be positive. The base must be positive and cannot equal 1. These rules come from how exponential functions behave.
A logarithm result can be a decimal. That simply means the needed exponent is between whole number powers of the base.
Change-of-base formula
A logarithm is the inverse of an exponent. If b^x = y, then log base b of y = x. The change of base formula lets you compute custom bases using any consistent log base.
Change of base: log_b(y) = log(y) / log(b). The log in the numerator and denominator must use the same base. It can be common log or natural log.
Log base 2 of 8
log base 10 of 1000 is 3 because 10^3 = 1000. ln(e^2) is 2 because ln uses base e.
For log base 2 of 32, ask what exponent turns 2 into 32. Since 2^5 = 32, the logarithm is 5.
For values between clean powers, expect a decimal result. log base 10 of 500 is between 2 and 3 because 500 is between 100 and 1000.
Invalid bases and values
The most common mistake is assuming log and ln always mean the same thing. Conventions can vary by course and field. Calculators commonly treat log as base 10 and ln as base e.
Another mistake is entering zero or a negative value. Real logarithms are only defined for positive values.
A third mistake is changing bases in the middle of a formula. If you use change of base, use the same log type in numerator and denominator.
Logarithm Calculator FAQ
What is the difference between log and ln?
ln is the natural logarithm and uses base e. On most calculators, log means base 10. In some advanced math settings, log may mean natural log if the instructor says so.
Follow the notation used by your problem, textbook or calculator.
How do I calculate a logarithm with a custom base?
Use the change of base formula. log_b(y) = log(y) / log(b). The logs in the numerator and denominator must use the same base.
For example, log base 2 of 8 can be calculated as log(8) / log(2), which equals 3.
This works with natural log too. ln(8) / ln(2) gives the same answer because both logs use the same underlying base.
Why must the log value be positive?
In real number math, a positive base raised to any real power never produces zero or a negative number. Because logarithms reverse exponents, the input must be positive.
Complex numbers can extend the idea, but that is outside ordinary calculator use.
Can the logarithm base be 1?
No. Base 1 is not allowed because 1 raised to any power is still 1. It cannot produce other positive values, so the inverse would not work.
The base also must be positive.
A valid real logarithm base is greater than 0 and not equal to 1. If the calculator rejects the base, check that rule before changing the value being logged.
What does a logarithm answer in plain English?
It answers, what exponent do I need? log base 10 of 100 asks what exponent turns 10 into 100. The answer is 2.
Reading logs this way makes the formula easier to remember.
This wording also helps you check reasonableness. If the base is 10 and the value is between 100 and 1000, the log should be between 2 and 3.
How can I check a logarithm answer?
Raise the base to the calculator result. If it returns the original value, the log result is consistent. For log base 2 of 8, the result is 3 because 2^3 is 8.
This check is especially helpful when using custom bases or change of base.