What is a natural logarithm?

A natural logarithm (ln) is a log with base e, where e is Euler's constant approximately equal to 2.7182818. It is used extensively in calculus, continuous compounding finance, and physics.

Why can log arguments not be negative or zero?

Because raising a positive base to any real power always results in a positive number (b^y > 0). Therefore, logs of negative numbers or zero are undefined in the real number system.

What is the change of base formula?

The change of base formula states that log_b(x) = ln(x) / ln(b). It allows you to compute logs of any base using natural or common log keys on a calculator.

Math Tool

Logarithm Calculator

Solve logarithms instantly. Supports natural logarithms (ln), common logarithms (log base 10), binary logarithms (log base 2), and any custom base.

Interactive Logarithm Calculator Workspace

Live Solver

Argument (Value x)

Calculated Logarithm Output2Equivalent statement: 10^2.0000 ≈ 100

Change of Base Proof Steps

Using the mathematical Change of Base formula: log_b(x) = ln(x) / ln(b)

Calculate Natural Log of Argument: ln(100) = 4.605170
Calculate Natural Log of Base: ln(10) = 2.302585
Divide terms: 4.605170 / 2.302585 = 2.000000

Mathematical Formula & Variables

Equation Modellog_b(x) = y => b^y = x

The logarithm of a number x with respect to base b is the exponent to which b must be raised to yield x. Natural logs use base e (≈2.71828).

Variable Definitions

Symbol	Description
b (Base)	The base of the logarithm (must be positive and not equal to 1)
x (Argument)	The value for which we want the exponent (must be greater than 0)

How to Use the Logarithm Calculator

Select your logarithm type: Common Log (log₁₀), Natural Log (ln), Binary Log (log₂), or Custom Base.
Enter the argument value (must be positive).
If using Custom Base, enter your desired base value.
The calculator will compute the log value, outline the algebraic equivalent power, and give steps.