# Logarithmic Functions

## Contents

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#### The Common Logarithmic Function

The function f (x) = log10 x is called the common logarithmic function. The common log function is often written as f (x) = log x -- when log is written without a base, the base is assumed to be 10. The " log " button on most calculators means log10 .

Examples:
log1000 = 3 .
log0.01 = - 2 .
log45 1.653 (using a calculator).

Figure %: f (x) = log x

#### The Natural Logarithmic Function

The number 2.71828… occurs often in mathematics and nature. For this reason, it is given a special name: e . Like Π , e is a number that mathematicians use often, and it is an irrational number, so it does not repeat or terminate. e has been calculated to many decimal places, but it is often rounded to e 2.718281828 .

 e 1 2.71828 e 2 7.38906 e 3 20.0855 e 4 54.5982 …

The function f (x) = loge x is called the natural logarithmic function. The natural log function is often abbreviated f (x) = ln x --this is the way it appears on most calculators. ln e x = x and e ln x = x .

Examples:
lne 4 = 4 .
ln = - 2 .
ln62 4.1271 .
ln230 5.4381 .
ln0.04 - 3.2189 .

Figure %: f (x) = ln x

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