Exponential functions and logarithms: Logarithmic functions
The logarithmic function
Earlier, we had a look at exponential functions. We'll now have a look at logarithms. The logarithm can be seen as the inverse, or opposite, of the exponential function. Let's take the function #\blue{a}^x#; it might be interesting to solve for #x# in the equation #\blue{a}^x=\green{b}#. When solving this equation, we'll use the logarithm.
Logarithm
The number #\log_{\blue{a}}\left(\green{b}\right)# is the exponent to which the number #\blue{a}# must be raised to get #\green{b}#. We call this the logarithm. Therefore
\[\begin{array}{lcr}\log_{\blue{a}}\left(\green{b}\right)=x&\text{ gives }&\blue{a}^x=\green{b}\end{array}\]
The number #\blue{a}# is called the base of the logarithm. The number #\blue{a}# has to be positive and not equal to #1#, the number #\green{b}# has to be positive.
Examples
\[\begin{array}{lcrl}\log_{\blue{2}}\left(\green{8}\right)&=&3&\text{because }\blue{2}^3=\green{8} \\ \log_{\blue{4}}\left(\green{\frac{1}{16}}\right)&=&-2&\text{because }\blue{4}^{-2}=\green{\frac{1}{16}} \\ \log_{\blue{5}}\left(\green{\sqrt{5}}\right)&=&\frac{1}{2}&\text{because }\blue{5}^{\frac{1}{2}}=\green{\sqrt{5}} \\ \end{array}\]
We can derive two important rules from the definition of logarithm.
\[\begin{array}{rcl}\blue{a}^{\log_{\blue{a}}\left(\green{b}\right)}&=&\green{b}\end{array}\]
Example
\[\begin{array}{lccr}\blue{3}^{\log_{\blue{3}}\left(\green{9}\right)}&=&\green{9}\end{array}\]
\[\begin{array}{rcl}\log_{\blue{a}}\left(\blue{a}^{\green{b}}\right)&=&\green{b}\end{array}\]
Example
\[\begin{array}{lccrr}\log_{\blue{a}}\left(1\right)&=&\log_{\blue{a}}\left(\blue{a}^\green{0}\right)&=&\green{0}\end{array}\]
# \begin{array}{rcl}
\log_{5}\left(25\right)&=&\log_{5}\left(5^2 \right)\\
&&\quad \blue{\text{identify }x\text{ such that }5^x=25}\\
&=& 2\\
&&\quad \blue{\log_{a}\left(a^b\right)=b}
\end{array} #
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