Glossary

Common logarithm

The common logarithm of a number x is the exponent to which we must raise 10 to obtain x. In other words, if 10y=x, then we say that y is the common logarithm of x, and we can write y=log10(x). We often abbreviate log10(x) to just log(x).

Because 10y is always positive, log(x) is only defined for positive x values. We can plot log(x) against x to obtain the function graph shown below. This function goes to as x approaches 0, and it goes off to (very, very slowly) as x tends to .

Common Logarithm

The logarithm has a number of properties that follow from its definition as an exponent:

  1. log10(xy)=log10(x)+log10(y);
  2. log10(1y)=log10(y);
  3. log10(xy)=log10(x)log10(y) (follows from (1) and (2));
  4. log10((x)n)=nlog10(x)

Other logarithm functions can be defined for other positive "bases". For example, the other most common logarithm is the natural logarithm: the natural logarithm of x, written loge(x) or ln(x), is the exponent to which we must raise e to obtain x (e=2.7182818284).

Any logarithm base will have a function graph having the same general shape as the common logarithm, and any logarithm base will enjoy analogous versions of the properties detailed above.

Wikipedia

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