Natural logarithm

Definition

The natural logarithm, denoted as $\ln(x)$, is the logarithm to the base $e$, where $e$ is an irrational constant approximately equal to 2.71828. It is the inverse function of the exponential function $e^x$.

5 Must Know Facts For Your Next Test

1. The natural logarithm function, $\ln(x)$, is defined only for positive real numbers ($x > 0$).
2. $\ln(1) = 0$ because $e^0 = 1$.
3. The derivative of $\ln(x)$ with respect to $x$ is $\frac{1}{x}$.
4. $\ln(e) = 1$ because $e^1 = e$.
5. For any positive real numbers $a$ and $b$, $\ln(ab) = \ln(a) + \ln(b)$ and $\ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b)$.

Review Questions

• What is the value of $\ln(1)$?
• Find the derivative of $\ln(x)$.
• Simplify $\ln(e^5)$.

Related terms

Exponential Function:

A function of the form $f(x) = e^x$, where $e$ is Euler's number (~2.71828).

$e$ (Euler's Number): An irrational constant approximately equal to 2.71828; it is the base of natural logarithms.

Logarithmic Properties: Rules such as $\log_b(mn) = \log_b(m) + \log_b(n)$ and $\log_b\left(\frac{m}{n}\right) = \log_b(m) - \log_b(n)$ that simplify logarithmic expressions.