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Math notes/Pre-Calculus/Logarithms

The Natural Logarithm — Cheat sheet

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The move

$\ln$ is a logarithm in base $e$ , where $e \approx 2.71828$ .

\ln x = \log_e x

Three notations

Written	Base
$\log_a x$	$a$ , stated
$\log x$	$10$ , hidden
$\ln x$	$e$ , hidden

The four rules with a = e

Rule	Why
$\ln 1 = 0$	$e^0 = 1$
$\ln e = 1$	$e^1 = e$
$\ln e^n = n$	the exponent is the answer
$e^{\ln x} = x$	the two cancel

Read at sight

$x$	$\ln x$
$e^7$	$7$
$e^2$	$2$
$e$	$1$
$1$	$0$
$\sqrt{e}$	$\dfrac{1}{2}$
$e^{1/3}$	$\dfrac{1}{3}$
$\dfrac{1}{e}$	$-1$
$\dfrac{1}{e^5}$	$-5$

Method

Write the argument as a power of $e$ .
Read off the exponent.
A root is a fractional power; a reciprocal is a negative one.

Watch out

$\ln$ has no $e$ written, but it is there. $\ln e^3 = 3$ , not $3e$ .
Domain unchanged: $\ln 0$ and $\ln(-4)$ have no value.
$\ln x$ is negative for $0 < x < 1$ , not undefined.
$e$ never has to be turned into a decimal to answer these.