Precalculus

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Rules of Thumb

The standard $\log$ with no base attached denotes $\log_{10}$
Logs cannot be negative by definition


What to Review

Logs from Algebra II
Definition of $e$


Properties

Of the form $a^{bx}$
Product rule: $\log_b(x\cdot y) = \log_b(x)+\log_b(y)$
Quotient rule: $\log_b(\frac{x}{y}) = \log_b(x)-\log_b(y)$
Power rule: $\log_b(x^y) = y\cdot\log_b(x)$
Base-Change rule: $\log_bx=\frac{\log_cx}{\log_cb}$


$e$

$e$ is a mathematical constant that is defined as $\displaystyle\lim_{n\to\infty}\left(1+\frac{1}{n}\right)^n$ or $\displaystyle\lim_{n\to0}(1+n)^\frac{1}{n}$
$e\approx2.71828$
$e$ can also be expressed as an infinite series: $\displaystyle\sum_{n=0}^\infty\frac{1}{n!}$
Logarithms with the base of $e$ are denoted as $\ln x$


Half-life Formula

$A=A_0(\frac{1}{2})^\frac{t}{h}$
$A$ is final amount of substance, $A_0$ is the initial amount, $t$ is time and $h$ is half-life