Log Calculator
Calculate logarithms using any positive base.
How Logarithms Are Calculated
A logarithm tells you the exponent required to raise a base to a given number. Logarithms are widely used in exponential growth, finance, acoustics, chemistry, computer science and scientific calculations.
Formula
logb(x) = ln(x) ÷ ln(b), where x must be positive and the base b must be positive and cannot equal 1.
Worked Example
For x = 1000 and base 10, log10(1000) = 3 because 10³ = 1000.
Factors That Affect the Result
The chosen base changes the numerical result. Base 10 is common for scientific notation, base e is used for natural logarithms and base 2 is frequently used in computing and information theory.
Frequently Asked Questions
What is the difference between log and ln?
Log often means base 10, while ln means the natural logarithm with base e.
Can the base be 1?
No. A logarithm with base 1 is undefined because 1 raised to any power is always 1.
Can I take the logarithm of a negative number?
Not in the real-number system. This calculator accepts only positive numbers.
Logarithm Rules and Domain
A logarithm answers the exponent required to produce a value from a chosen base. For real-number calculations, the input must be positive and the base must be positive and not equal to 1. Common logarithms use base 10, while natural logarithms use base e. Keep the same base when applying logarithm identities.
Methodology, Assumptions and Interpretation
A logarithm answers which exponent produces a value: log_b(x)=y means b^y=x. For real-valued logs, the input must be positive and the base must be positive and not equal to 1; base 10 and natural log base e are common special cases.