# Exponential and Logarithmic Functions

 
 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 $b$. $\log_a x = \dfrac{\log_b x}{\log_b a}$