Question 171046

The appropriate logarithm rules to apply are:


1. {{{log(b,x[1])+log(b,x[2])=log(b,(x[1]*x[2]))}}} and


2. {{{log(b,x[1])-log(b,x[2])=log(b,(x[1]/x[2]))}}}


First thing is to apply the Associative and Distributive Properties:


{{{log(3)-log(4)-log(7)=log(3)-(log(4)+log(7))}}}


Now, using Rule 1., you can render the part in parentheses as:


{{{log(28)}}} and substitute, thus:


{{{log(3)-log(28)}}}


Now apply rule 2., thus:


{{{log(3/28)}}}


Since {{{3/28}}} is already reduced to lowest terms, you are done.


Check the reasonability of your answer with a calculator:


{{{log(3) = 0.4771}}} (roughly)


{{{log(4) = 0.6021}}} (roughly)


{{{log(7) = 0.8451}}} (roughly)


{{{0.4771 - 0.6021 = -0.125}}} and {{{-0.125 - 0.8451 = -0.9701}}}


So far, so good.  Since {{{log(10,1)=0}}}, the log of something smaller than 1, in particular {{{3/28}}} must be less than zero.


Again, from the calculator:  {{{3/28 = 0.1071}}} (roughly) and {{{log(0.1071) = -0.9072}}}.


Close enough.