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