prove that log 8 to the base 8 divided by log 16 to the base 9× log 10 to the base 4 = 3log 2 to the base 10
Answers
Answer:
log 8 base 2+log 8 base 4+log 8base 16.
By change of bases, we can write log a base b as log a base x divided by log b base x.
log 8 base 2 can be written as log 2^3 base 2 divided by log 2 base 2 ...(*)
log 8 base 4 = log 2^3 base 2 divided by log 2^2 base 2 ...(**)
log 8 base 16 = log 2^3 base 2 divided by log 2^4 base 2 ...(***)
By the rule, log a ^m = m log a,
(*) can be written as 3log 2 base 2 divided by log 2 base 2 ..(****)
(**) can be written as 3log 2 base 2 divided by 2 log 2 base 2 ..(*****)
(***) can be written as 3log 2 base 2 divided by 4 log 2 base 2 ..(*****)
Taking 3 log 2 base 2 as common,
3 log 2 base 2{ (1/log 2 base 2 )+(1/2 log 2 base 2 )+(1/ 4 log 2 base 2 )}
we know that log a base a is equal to 1.
therefore, 3{ 1 + (1/2)+(1/4)}
on simplifying, we get,21/4