SOLUTION: Bear with me I'm not sure how to enter certain symbols:( Sorry.) Expand as much as possible and evaluate: log3(base) then radicand symbol (index is 4) and x^5 y^4/81 is inside

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Question 116270: Bear with me I'm not sure how to enter certain symbols:( Sorry.)
Expand as much as possible and evaluate:
log3(base) then radicand symbol (index is 4) and x^5 y^4/81 is inside the
radicand symbol.
I noticed the two equations inside the radicand symbol were divided so I
know this means I have to write it as subtraction.
log3(x^5y^4)^1/4-log3 (81)1/4
log3 (x^5/4 y^4/4) - 1/4 log3 (81)?
5/4 log3 X + 4/4 log3 Y - 9/4?
Please help our study group was coming up with 3 different answers and I'm not
sure any of us were sure what to do.:)
Found 3 solutions by psbhowmick, jim_thompson5910, bucky:
Answer by psbhowmick(489) About Me  (Show Source):
You can put this solution on YOUR website!
log%283%2C%28x%5E5y%5E4%2F81%29%5E%281%2F4%29%29
= +log%283%2Cx%5E%285%2F4%29%29+%2B+log%283%2Cy%5E%284%2F4%29%29+-+log%283%2C81%5E%281%2F4%29%29+
= +%285%2F4%29log%283%2Cx%29+%2B+log%283%2Cy%29+-+log%283%2C%283%5E4%29%5E%281%2F4%29%29+
= +%285%2F4%29log%283%2Cx%29+%2B+log%283%2Cy%29+-+log%283%2C3%29+
= +%285%2F4%29log%283%2Cx%29+%2B+log%283%2Cy%29+-+1+
Hope this helps!

Answer by jim_thompson5910(21667) About Me  (Show Source):
You can put this solution on YOUR website!
log%283%2C%28root%284%2C%28x%5E5y%5E4%29%2F81%29%29%29 Start with the given expression


log%283%2C%28%28%28x%5E5y%5E4%29%2F81%29%5E%281%2F4%29%29%29 Rewrite root%284%2C%28x%5E5y%5E4%29%2F81%29 as %28%28x%5E5y%5E4%29%2F81%29%5E%281%2F4%29


log%283%2C%28%28%28x%5E%285%2F4%29y%5E%284%2F4%29%29%2F81%5E%281%2F4%29%29%29%29 Distribute the exponent


log%283%2C%28%28%28x%5E%285%2F4%29y%29%2F3%29%29%29 Simplify


log%283%2C%28%28x%5E%285%2F4%29y%29%29%29-log%283%2C%283%29%29 Break up the log using the identity log%28b%2C%28x%2Fy%29%29=log%28b%2C%28x%29%29-log%28b%2C%28y%29%29


log%283%2C%28%28x%5E%285%2F4%29%29%29%29%2Blog%283%2C%28y%29%29-log%283%2C%283%29%29 Break up the log using the identity log%28b%2C%28x%2Ay%29%29=log%28b%2C%28x%29%29%2Blog%28b%2C%28y%29%29


%285%2F4%29log%283%2C%28x%29%29%2Blog%283%2C%28y%29%29-log%283%2C%283%29%29 Rewrite the first log using the identity log%28b%2C%28x%5Ey%29%29=y%2Alog%28b%2C%28x%29%29


%285%2F4%29log%283%2C%28x%29%29%2Blog%283%2C%28y%29%29-1 Now evaluate log%283%2C%283%29%29 to get 1

Answer by bucky(2097) About Me  (Show Source):
You can put this solution on YOUR website!
Given:
.
log%283%2Croot%284%2C%28x%5E5%2Ay%5E4%29%2F81%29%29
.
Your work was:
.
log%283%2C%28%28x%5E5%2Ay%5E4%29%5E%281%2F4%29%29%29-log%283%2C%28%2881%29%5E%281%2F4%29%29%29 <=== ok
.
log%283%2C+%28x%5E%285%2F4%29%29%2A%28+y%5E%284%2F4%29%29%29+-+%281%2F4%29%2Alog%283%2C81%29<=== ok
.
%285%2F4%29%2A+log%283%2Cx%29+%2B+%284%2F4%29%2A+log%283%2Cy%29+-+9%2F4 <=== mistake in last term
.
You have a mistake in evaluating %28-1%2F4%29+%2Alog%283%2C81%29. Replace 81 with 3%5E4%29 and the term
becomes:
.
%28-1%2F4%29+%2Alog%283%2C3%5E4%29
.
Bring the exponent 4 out as a multiplier and the result is:
.
%28-1%2F4%29%2A4+%2Alog%283%2C3%29
.
But log%283%2C3%29+=+1
.
so this reduces to:
.
%28-1%2F4%29+%2A+4+%2A+1+=+-1
.
So in your last equation replace - 9/4 with -1 and you have:
.
%285%2F4%29%2A+log%283%2Cx%29+%2B+%284%2F4%29%2A+log%283%2Cy%29+-+1
.
and in the second term note that 4%2F4+=+1 so the expression reduces to:
.
%285%2F4%29%2A+log%283%2Cx%29+%2B+log%283%2Cy%29+-+1
.
That should be enough to convince your teacher that you know what you are doing. I'm not sure
what else could be productively done on this problem. If you feel that it needs more work,
post it again with the above change to your work and see if some other tutor has another idea
about how to go further.
.
Hope this helps you and the study group. Check my work and ensure that I haven't made a "late night"
error.
.