Expert: Paul Klarreich - 10/13/2008

Question
hi

Question:

Express log(2root10) -1/3log(0.8) -log(10/3) in the form c+logd where c and d are rational numbers and the logarithms are to the base 10

answer is log3-1/6 ...........I have the log3 and some derived 1s but am struggling with the -1/3 log (0.8) part of the question.

Answer
Questioner:   rg
Category:  Advanced Math
Private:  No
 
Subject:  express log ...... in the form c + log d
Question:  hi

Question:

Express log(2root10) -1/3log(0.8) -log(10/3) in the form c+logd where c and d are rational numbers and the logarithms are to the base 10

answer is log3-1/6 ...........I have the log3 and some derived 1s but am struggling with the -1/3 log (0.8) part of the question.
.................................
Hi, rg,

If the logs are using base 10, then look for instances of  log(10), which is equal to 1.  For example:

log (2 sqrt(10)) = log 2 + log(sqrt(10) = log 2 + 1/2 log 10 = log 2 + 1/2

or

log (0.8) = log (8/10) = log 8 - log 10 = log(2^3) - 1 = 3 log 2 - 1

or

log (10/3) = log 10 - log 3 = 1 - log 3

Now put it all together:

log(2 sqrt(10)) - 1/3 log(0.8) - log(10/3) =

log 2 + 1/2 - 1/3(3 log 2 - 1) - (1 - log 3)

log 2 + 1/2 - log 2 + 1/3 - 1 + log 3

     + 1/2         + 1/3 - 1 + log 3

     + 5/6               - 1 + log 3

     - 1/6 + log 3

That's it.