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 $e^x$.
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The natural logarithm function, $\ln(x)$, is defined only for positive real numbers ($x > 0$).
$\ln(1) = 0$ because $e^0 = 1$.
The derivative of $\ln(x)$ with respect to $x$ is $\frac{1}{x}$.
$\ln(e) = 1$ because $e^1 = e$.
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.