# Logarithmic Functions

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## Logarithmic Function

 The logarithmic function of base a, where a is positive and not 1, is denoted by ${\displaystyle y=log_{a}(x)}$
(which is read as "y is log base a of x") and is defined by
${\displaystyle y=log_{a}(x){\text{ if and only if }}a^{y}=x}$


## Properties

 Domain of logarithmic function = range of exponential function = ${\displaystyle (0,\infty )}$
Range of logarithmic function = domain of exponential function = ${\displaystyle (-\infty ,\infty )}$


In fact the logarithmic function ${\displaystyle f(x)=log_{a}(x)}$ is the inverse of ${\displaystyle g(x)=a^{x}}$

## Properties of the graph

 Properties of ${\displaystyle f(x)=log_{a}(x),~a>1,a\neq 1}$
1. The domain is ${\displaystyle (0,\infty )}$ and the range is ${\displaystyle (-\infty ,\infty )}$
2. The x-intercept is (1, 0) and there is no y-intercept.
3. The y-axis is a horizontal asymptote
4. ${\displaystyle f(x)}$ is an increasing if ${\displaystyle a>1}$ and decreasing if ${\displaystyle 0
5. one-to-one function
6. The graph contains the three points ${\displaystyle (1,0),~(a,1),~({\frac {1}{a}},-1)}$
7. The graph of f is smooth and continuous. (Here smooth means you can take as many derivatives as you want)


## Common Logarithm

Sometimes a logarithm function is written without making reference to a base, for example ${\displaystyle f(x)=\log(x)}$

When this happens the base is assumed to be 10. This means ${\displaystyle \log(x)=\log _{10}(x)}$

## Natural Logarithm

 There is a special base, e, to which we associate a special logarithm ${\displaystyle \ln }$, which is called the natural logarithm.

 ${\displaystyle y=\ln(x){\text{if and only if}}e^{x}=y}$


Notice that we do not write the base. That is whenever we use the natural logarithm, we are using base e.

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