Logarithmic Functions

Logarithmic Function

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

Properties

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

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

Properties of the graph

 Properties of $f(x)=log_{a}(x),~a>1,a\neq 1$
 1. The domain is $(0,\infty )$ and the range is $(-\infty ,\infty )$
 2. The x-intercept is (1, 0) and there is no y-intercept.
 3. The y-axis is a horizontal asymptote
 4. $f(x)$ is an increasing if $a>1$ and decreasing if $0<a<1$
 5. one-to-one function
 6. The graph contains the three points $(1,0),~(a,1),~({\frac {1}{a}},-1)$
 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 $f(x)=\log(x)$

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

Natural Logarithm

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

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

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

Note: e is about 2.71828...

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