SOLUTION: Given that log2 3=x, log2 5 = y,and log2 7=z,express logx square root 21 in terms of x, y, z. Show work please.

Algebra ->  Logarithm Solvers, Trainers and Word Problems -> SOLUTION: Given that log2 3=x, log2 5 = y,and log2 7=z,express logx square root 21 in terms of x, y, z. Show work please.      Log On


   

Question 1206896: Given that log2 3=x, log2 5 = y,and log2 7=z,express logx square root 21 in terms of x, y, z. Show work please.
Found 5 solutions by greenestamps, MathLover1, Theo, ikleyn, Edwin McCravy:
Answer by greenestamps(13195) About Me  (Show Source):
You can put this solution on YOUR website!


I will guess that the expression is supposed to be log2 square root 21 and not logx square root of 21....



ANSWER: %28x%2Bz%29%2F2

Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!
Given that

log%282%2C+3%29=x

log%282+%2C5%29+=+y

log%282+%2C7%29=z

express log%282%2Csqrt%28+21%29%29 in terms of+x, y,+z

log%282%2Csqrt%28+21%29%29

=log%282%2Csqrt%28+3%2A7%29%29

=log%282%2C%28+3%2A7%29%5E%281%2F2%29%29

=%281%2F2%29log%282%2C%28+3%2A7%29%29

=%281%2F2%29%28log%282%2C3%29%2Blog%282%2C7%29%29....substitute given

=%281%2F2%29%28x%2Bz%29



Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!
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.



log%28x%2Csqrt%2821%29%29%22%22=%22%22log%282%2Csqrt%2821%29%29%2Flog%282%2Cx%29%22%22=%22%22log%282%2C21%5E%28%221%2F2%22%29%29%2Flog%282%2Cx%29%22%22=%22%22%28expr%281%2F2%29log%282%2C21%29%29%2Flog%282%2Cx%29%22%22=%22%22log%282%2C21%29%2F%282%2Alog%282%2Cx%29%29%22%22=%22%22log%282%2C%283%2A7%29%29%2F%282%2Alog%282%2Cx%29%29%22%22=%22%22
%28log%282%2C3%29%2Blog%282%2C7%29%29%2F%282%2Alog%282%2Cx%29%29%22%22=%22%22%28x%2Bz%29%2F%282%2Alog%282%2Cx%29%29

To get y into the picture we look at

log%282%2C5%29=y
2%5Ey=5
%282%5Ey%29%5E%28%221%2Fy%22%29=5%5E%28%221%2Fy%22%29
2=5%5E%28%221%2Fy%22%29

So we substitute that for the 2 in the denominator of

%28x%2Bz%29%2F%282%2Alog%282%2Cx%29%29
and the final answer is
%28x%2Bz%29%2F%285%5E%28%221%2Fy%22%29%2Alog%282%2Cx%29%29%29

Edwin


THANK YOU EDWIN, THAT WAS BEAUTIFULLY DONE.


Answer by ikleyn(52756) About Me  (Show Source):
You can put this solution on YOUR website!
.

comment from Edwin : The solution was to express log%28x%2Csqrt%2821%29%29 in terms of x,y, and z.
The solution is: %28x%2Bz%29%2F%285%5E%28%221%2Fy%22%29%2Alog%282%2Cx%29%29%29 Edwin


My response: Edwin, I trace very attentively your posts, as well as the posts of all other tutors.

It did not escape my attention that you solved this problem.

And I am very impressed of it.


Thank you for posting your comment to me.


                                                BUT.


But I have my own ideas/suggestions what is a correctly posed Math problem and what is a good solution to a Math problem.

From my point of view, this problem is posed incorrectly, is not good and unlikely to have a good solution.

Concretely, to me this problem is like a crooked sidewalk that's best avoided.



Answer by Edwin McCravy(20054) About Me  (Show Source):
You can put this solution on YOUR website!


log%28x%2Csqrt%2821%29%29%22%22=%22%22log%282%2Csqrt%2821%29%29%2Flog%282%2Cx%29%22%22=%22%22log%282%2C21%5E%28%221%2F2%22%29%29%2Flog%282%2Cx%29%22%22=%22%22%28expr%281%2F2%29log%282%2C21%29%29%2Flog%282%2Cx%29%22%22=%22%22log%282%2C21%29%2F%282%2Alog%282%2Cx%29%29%22%22=%22%22log%282%2C%283%2A7%29%29%2F%282%2Alog%282%2Cx%29%29%22%22=%22%22
%28log%282%2C3%29%2Blog%282%2C7%29%29%2F%282%2Alog%282%2Cx%29%29%22%22=%22%22%28x%2Bz%29%2F%282%2Alog%282%2Cx%29%29

To get y into the picture we look at

log%282%2C5%29=y
2%5Ey=5
%282%5Ey%29%5E%28%221%2Fy%22%29=5%5E%28%221%2Fy%22%29
2=5%5E%28%221%2Fy%22%29

So we substitute that for the 2 in the denominator of

%28x%2Bz%29%2F%282%2Alog%282%2Cx%29%29
and the final answer is
%28x%2Bz%29%2F%285%5E%28%221%2Fy%22%29%2Alog%282%2Cx%29%29%29

Edwin