SOLUTION: Hello there, For the following problem, I am not sure if my answer is right or if the given answer is right. In the case that the given answer is correct, could someone please e

Algebra ->  Exponential-and-logarithmic-functions -> SOLUTION: Hello there, For the following problem, I am not sure if my answer is right or if the given answer is right. In the case that the given answer is correct, could someone please e      Log On


   

Question 118399: Hello there,
For the following problem, I am not sure if my answer is right or if the given answer is right. In the case that the given answer is correct, could someone please explain where I might have erred?
Thank you !
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1. +%28log%28+3%2C+2+%29%29%28log+%282%2C+3%29+sqrt%283%29%29%29+
= +%281%2F%28log+%282%2C+3+%29%29%29%28log+%282%2C+3%29+sqrt%28+3+%29%29%29%29+
= +%28log+%282%2C+3%29+sqrt+%283%29%29%2F%28log+%282%2C+3%29%29+
= +sqrt%283%29+
However, the given answer is +%28log+%282%2C+%281%2F3%29%29%29+.
Found 2 solutions by jim_thompson5910, ilana:
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
+%28log%28+3%2C+%282%29+%29%29%28log+%282%2C+%283%2Asqrt%283%29%29%29%29+ Start with the given expression


Rewrite the logs using the change of base formula



Cancel like terms


+%281%2Flog%2810%2C3%29%29%28log%2810%2C3%2Asqrt%283%29%29%2F1%29+ Simplify


log%2810%2C3%2Asqrt%283%29%29%2Flog%2810%2C3%29 Combine the fractions



%28log%2810%2C3%29%2Blog%2810%2Csqrt%283%29%29%29%2Flog%2810%2C3%29 Break up the log using the identity log%28b%2CA%2AB%29=log%28b%2CA%29%2Blog%28b%2CB%29


log%2810%2C3%29%2Flog%2810%2C3%29%2Blog%2810%2Csqrt%283%29%29%2Flog%2810%2C3%29 Break up the fraction


cross%28log%2810%2C3%29%2Flog%2810%2C3%29%29%2Blog%2810%2Csqrt%283%29%29%2Flog%2810%2C3%29 Cancel like terms


1%2Blog%2810%2Csqrt%283%29%29%2Flog%2810%2C3%29 Simplify


1%2Blog%2810%2C3%5E%281%2F2%29%29%2Flog%2810%2C3%29 Rewrite sqrt%283%29 as 3%5E%281%2F2%29


1%2B%281%2F2%29log%2810%2C3%29%2Flog%2810%2C3%29 Rewrite the log using the identity log%28b%2Cx%5Ey%29=y%2Alog%28b%2Cx%29


1%2B%281%2F2%29cross%28log%2810%2C3%29%2Flog%2810%2C3%29%29 Cancel like terms


1%2B1%2F2 Simplify



3%2F2 Add the fractions






So +%28log%28+3%2C+%282%29+%29%29%28log+%282%2C+%283%2Asqrt%283%29%29%29%29=3%2F2+


I checked this answer with my calculator, so you must have typed in the problem incorrectly or it might be displayed incorrectly. Double check your book to verify that you have the right problem.

Answer by ilana(307) About Me  (Show Source):
You can put this solution on YOUR website!
It seems to me you are both wrong. Your last step does not work. Think of it this way: log(base2)8/log(base2)4=3/2, but 8/4=2. So just dividing does not work. To check your answers, I used the fact that log(base3)2 = log2/log3 (with any base- so the calculator uses base 10). So I did (log2/log3)*(log3rt3/log2)=log3rt3/log3=1.5
I honestly do not remember any rule that would help me get that without my calculator, but I know that the rules I used are true, so I believe the answer is 1/3.