Natural logarithm

Definition

The natural logarithm is the logarithm to the base $e$, where $e$ is an irrational and transcendental number approximately equal to 2.71828. It is commonly denoted as $\ln(x)$.

5 Must Know Facts For Your Next Test

1. The natural logarithm of a positive number $x$ is written as $\ln(x)$.
2. The base $e$ of the natural logarithm is approximately equal to 2.71828.
3. $\ln(e) = 1$ and $\ln(1) = 0$ are key properties.
4. The derivative of $\ln(x)$ with respect to $x$ is $\frac{1}{x}$.
5. The natural logarithm function is the inverse of the exponential function $e^x$.

Review Questions

• What is the value of $\ln(e)$?
• What does the natural logarithm function represent?
• How do you express the natural logarithm of a product using properties of logarithms?

Related terms

Exponential Function:

A mathematical function in the form $f(x) = e^x$, where $e$ is Euler's number.

Logarithmic Function:

A function that uses a logarithm, typically written as $\log_b(x)$ for a given base $b$. The natural log specifically uses base $e$.

Euler's Number:

An irrational and transcendental number approximately equal to 2.71828, often denoted as $e$. It serves as the base for natural logarithms.