Chapter 6 Elementary Functions

Section 6.4 Exponential and Logarithmic Functions

6.4.5 Logarithm Rules


For calculations involving logarithmic functions certain rules apply that can be derived form the exponent rules.
Info 6.4.11
 
The following rules are called logarithm rules:

log(u·v)=log(u)+log(v)(u,v>0) , log( u v )=log(u)-log(v)(u,v>0) , log( ux )=x·log(u)(u>0,x∈ℝ) .


These rules do not only apply to natural logarithmic functions but also to all other logarithmic functions. They can be used to transform a given expression in such a way that the power occurs only in the logarithmic terms.
Example 6.4.12
For example, the value ld ( 45 ) can be calculated applying the logarithm rules:

ld ( 85 )  =   log2 ( 85 )  =  5· log2 (8)  =  5· log2 ( 23 )  =  5·3  =  15 .

Products in logarithmic functions can be split into sums outside the logarithmic functions:

lg (100·10· 1 10 )  =   lg (100)+ lg (10)- lg (10)  =  2+ 1 2 -1  =   3 2  .


Importantly, the splitting rule log(u·v)=log(u)+log(v) transforms products into sums. The other way round is impossible for logarithmic functions: the logarithm of a sum cannot be transformed any further.
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