Relationship Between Exponentials and Logarithms
In what follows, $x$ and $y$ are real numbers, $a$ and $b$ are positive real numbers not equal to $1$.
-
$\log_a x = y$ if and only if $x = a^y$
- $\log_a (a^x) = x$
- $a^{\log_a x} = x$ for $x \gt 0$
Laws of Exponents
In what follows, $x$ and $y$ are real numbers, $a$ and $b$ are positive real numbers.
- $a^x \cdot a^y=a^{x+y}$
- $\dfrac{a^x}{a^y}=a^{x-y}$
- $(a^x)^y=a^{x \cdot y}$
- $a^{-x}=\dfrac{1}{a^x}$
- $(a \cdot b)^x = a^x \cdot b^x$
- $a^0=1$
Exponents and Radicals
In what follows, $m$ and $n$ are positive integers.
- $\sqrt[n]a = a^{\tfrac{1}{n}}$
- $\sqrt[n]{a^m} = a^{\tfrac{m}{n}}$
- $\sqrt[n]{\dfrac{a}{b}} = {\dfrac{\sqrt[n]a}{\sqrt[n]b}}$
Laws of Logarithms
In what follows, $W$ and $U$ are positive real numbers, $a$ and $b$ are positive real numbers not equal to $1$.
- $\log_a 1 = 0$
- $\log_a W \cdot U = \log_a W + \log_a U$
- $\log_a \dfrac {W}{U}= \log_a W - \log_a U$
- $\log_a {W ^x}= x \log_a W$
- $\log_a \dfrac {1}{U}= - \log_a U$
Change of Base Formula for Logarithms
A formula to change the base $a$ of a logarithm to base $b$.
- $\log_a x = \dfrac{\log_b x}{\log_b a}$
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