Logarithmic functions > Properties
12345Properties

Theory

Definition of a logarithm:
g x = y is equivalent to x = g log ( y ) if 0 < g < 1 or g > 1 and if y > 0 .

Definition formulas:
It follows from the definition of a logarithm that: g log ( g x ) = x and g g log ( y ) = y .

Properties of logarithms:
If 0 < g < 1 or g > 1 and if a > 0 and b > 0 the following properties hold:

  • g log ( a ) + g log ( b ) = g log ( a b )

  • g log ( a ) g log ( b ) = g log ( a b )

  • p g log ( a ) = g log ( a p )

Changing the base:
In order to be able to work with a desired base (the log-button of your calculator always uses 10) you need to be able to change an existing base.
From the properties of logarithms you can derive: g log ( a ) = p log ( a ) / p log ( g ) .
In this way you can determine logarithms with your calculator and/or enter them as a function. Note that modern calculators sometimes allow you to choose the base of the logarithm. You then need to use the American notation log g ( x ) , where the base is written in a different position.

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