Properties of logarithms, Algebra

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Properties of Logarithms

1. logb1 = 0 .  It follows from the fact that bo  = 1.

2. logb b = 1.  It follows from the fact that b 1= b .

3. logb bx  = x .  it can be generalized out to blog b f ( x )  = f ( x ).

4. b logb x  = x .  It can be generalized out to b logb  f ( x )  = f ( x ) .

Properties 3 and 4 lead to a pleasant relationship among the logarithm & exponential function.

Let's first calculate the following function compositions for f ( x )= b x and g ( x ) = logb x .

 ( f o g )( x ) = f [g ( x )] =  f (logb x ) = b logb x  = x

 ( g o f ) ( x ) = g [f ( x )]= g [b x ] = log b bx  = x

Remember again from the section on inverse functions which this means that the exponential & logarithm functions are inverses of each other. It is a nice fact to remember on occasion.

We have to also give the generalized version of Properties 3 & 4 in terms of both the natural and common logarithm

ln e f ( x )  = f ( x)                               log10 f ( x )  = f ( x)

eln f ( x )  = f ( x )                                    10log f ( x )  = f ( x )


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