Logarithm Calculator

A logarithm is the inverse of an exponent. It answers the question: How many times do we need to multiply a base number by itself to get another specific number? For example, the logarithm base 10 of 100 is 2 because 10 multiplied by itself 2 times (10 × 10) equals 100.

The main difference is the base number. A regular log (usually written as 'log') typically uses base 10, which matches our everyday decimal number system. The natural log (written as 'ln') uses Euler's number 'e' as its base, which is approximately 2.71828. We use 'ln' heavily in physics, economics, and compound growth formulas.

Logarithms ask how many times we multiply a positive base by itself to get a target number. Multiplying positive numbers together will always result in a positive number. There is no way to multiply a positive base to get zero or a negative value. Therefore, logs for numbers less than or equal to zero are mathematically undefined for real numbers.

You can solve logs with any custom base by using the change-of-base formula. This formula states that log base b of x is equal to the natural log of x divided by the natural log of b (ln(x) / ln(b)). You can also use log base 10 (log10(x) / log10(b)). Our calculator runs this division instantly behind the scenes.

Euler's number (e ≈ 2.71828) is a mathematical constant representing the base of natural growth. It occurs naturally in situations involving continuous compounding interest, population growth rates, and radioactive decay patterns.