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# Introduction to Logarithms

### Logarithm rules

• Equivalence of logarithms and power notation:
\begin{eqnarray*}
a^{x}= y &\leftrightarrow & \log_{a} y = x \\
e^{x}=y &\leftrightarrow& \ln y = x
\end{eqnarray*}
• Logarithm of a product rule:
$\log_{a} (u\times v) = \log_{a}(u) +\log_{a}(v)$
• Logarithm of a quotient or division rule:
$\log_{a}\left( \frac{u}{v}\right) = \log_{a}(u) - \log_{a}(v)$
• Logarithm of a power rule:
$\log_a x^n = n \log_a x$
• Logarithm of number to the same base $$\log_a a = 1$$
• Logarithm of $$1$$ to any base: $$\log_a 1 = 0$$

Examples:

• $$\log_{3} 2 + \log_{3} x = \log_{3}(2\times x) = \log_{3} (2x)$$
• $$\log_{3} 2 - \log_{3} x = \log_{3} \left(\frac{2}{x}\right)$$
• Simplifying
\begin{eqnarray*}
3 \log 2 + \log 125&=& \log 2^3 + \log 125 \\
&=& \log 8 + \log 125 \\
&=& \log (8\times 125) \\
&=& \log(1000) \\
&=& 3
\end{eqnarray*}
• Simplifying
\begin{eqnarray*}
\log xy^3 - \log x^2y + \log x
&=& \log \left( \frac{xy^3}{x^2y}\times x \right) \\
&=& \log y^2 \\
&=& 2 \log y
\end{eqnarray*}