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==Logarithmic Function== The logarithmic function of base a, where a is positive and not 1, is denoted by $y = log_a(x)$ (..."

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==Logarithmic Function==

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

==Properties==

Domain of logarithmic function = range of exponential function = <math>(0, \infty)</math>
Range of logarithmic function = domain of exponential function = <math>(-\infty, \infty)</math>

In fact the logarithmic function <math> f(x) = log_a(x) </math> is the inverse of <math> g(x) = a^x</math>

==Properties of the graph==

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

When this happens the base is assumed to be 10. This means <math> \log(x) = \log_{10}(x)</math>

==Natural Logarithm==

There is a special base, e, to which we associate a special logarithm <math> \ln</math>, which is called the natural logarithm.

<math> y = \ln(x) \text{if and only if} e^x = y</math>

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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Logarithmic Functions - Revision history

MathAdmin: Created page with "__TOC__ ==Logarithmic Function== The logarithmic function of base a, where a is positive and not 1, is denoted by y = log_a(x) (..."

MathAdmin: Created page with "
TOC
==Logarithmic Function== The logarithmic function of base a, where a is positive and not 1, is denoted by $y = log_a(x)$ (..."