i get:
log2(3) = log(3)/log(2) = 1.584962501.
by log rules, log2(3) = y if and only if 2^y = 3
when y = 1.584962501, you get 2^1.584962501 = 3.
use your calculator to confirm this is true.
likewise, .....
log2(5) = log(5)/log(2) = 2.321928095.
log2(7) = log(7)/log(2) = 2.807354922.
you want log2(sqrt(21).
that's the same as log2(sqrt(3*7).
that's the same as log2((3*7)^.5).
that's the same as .5 * log2(3*7).
that's the same as .5 * (log2(3) + log2(7)).
since log2(3) = x and log2(7) = z, that's the same as ...
log2(sqrt(21) = .5 * (x + z).
to confirm, use the fact that log2(x) = log(x)/log(2).
using your calculator, find that log2(sqrt(21)) = log(sqrt(21))/log(2) = 2.196158711.
using your calculator, find that .5 * (x + z) = .5 * (log2(3) + log2(7)) = .5 * (log(3)/log(2) + log(7)/log(2)) = 2.1961858711.
this means they're equivalent.
i didn't see where log2(5) played into this and i didn't understand what you meant by logx(sqrt(21)).
THIS SOLUTION HAS BEEN CORRECTED BY EDWIN MCRAVEY AS SHOWN BELOW.
All the other tutors misinterpreted your problem. They thought x
couldn't be the base of a log, but there is no reason why it can't.
To get y into the picture we look at
So we substitute that for the 2 in the denominator of
and the final answer is
Edwin
THANK YOU EDWIN, THAT WAS BEAUTIFULLY DONE.