Exponential and Logarithmic Functions

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Relationship Between Exponentials and Logarithms

    In what follows, $x$ and $y$ are real numbers, $a$ and $b$ are positive real numbers not equal to $1$.

  1. $\log_a x = y$ if and only if $x = a^y$

  2. $\log_a (a^x) = x$

  3. $a^{\log_a x} = x$ for $x \gt 0$

Laws of Exponents

    In what follows, $x$ and $y$ are real numbers, $a$ and $b$ are positive real numbers.

  1. $a^x \cdot a^y=a^{x+y}$

  2. $\dfrac{a^x}{a^y}=a^{x-y}$

  3. $(a^x)^y=a^{x \cdot y}$

  4. $a^{-x}=\dfrac{1}{a^x}$

  5. $(a \cdot b)^x = a^x \cdot b^x$

  6. $a^0=1$

Exponents and Radicals

    In what follows, $m$ and $n$ are positive integers.

  1. $\sqrt[n]a = a^{\tfrac{1}{n}}$

  2. $\sqrt[n]{a^m} = a^{\tfrac{m}{n}}$

  3. $\sqrt[n]{\dfrac{a}{b}} = {\dfrac{\sqrt[n]a}{\sqrt[n]b}}$

Laws of Logarithms

    In what follows, $W$ and $U$ are positive real numbers, $a$ and $b$ are positive real numbers not equal to $1$.

  1. $\log_a 1 = 0$

  2. $\log_a W \cdot U = \log_a W + \log_a U$

  3. $\log_a \dfrac {W}{U}= \log_a W - \log_a U$

  4. $\log_a {W ^x}= x \log_a W$

  5. $\log_a \dfrac {1}{U}= - \log_a U$

Change of Base Formula for Logarithms

    A formula to change the base $a$ of a logarithm to base $b$.

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