SOLUTION: Which one has more value(or which is bigger?)? log2(3) or log3(8) Solve it without using calculator,please.

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Question 955354: Which one has more value(or which is bigger?)? log2(3) or log3(8)
Solve it without using calculator,please.
Found 3 solutions by MathLover1, herosz, ikleyn:
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

log%282%2C%283%29%29 or log%283%2C%288%29%29.......change the base
log%28%283%29%29%2Flog%282%29 or log%28%288%29%29%2Flog%283%29
log%28%283%29%29%2Flog%282%29 or log%28%282%5E3%29%29%2Flog%283%29
log%28%283%29%29%2Flog%282%29 or 3log%28%282%29%29%2Flog%283%29
so,
log%283%2C%288%29%29%3Elog%282%2C%283%29%29

Answer by herosz(4) About Me  (Show Source):
You can put this solution on YOUR website!
Unfortunately MathLover1's answer is not self-evident!
for example assuming log3=2/3 and log2=1/3 (we don't use calculator so don't know the exact value) ==>
(2/3)*3>(3*1/3)*3/2 so log2(3)>log3(8) that is not true!

Answer by ikleyn(52795) About Me  (Show Source):
You can put this solution on YOUR website!
.
Which number is greater: log%282%2C%283%29%29 or log%283%2C%288%29%29 ?
~~~~~~~~~~~~~~~~~ 

We have, from one side,

4%2Alog%282%2C%283%29%29 = log%282%2C%283%5E4%29%29 = log%282%2C%2881%29%29 < log%282%2C%28128%29%29 = 7.   Thus  log%282%2C%283%29%29 < 7%2F4.



We have, from the other side,

4%2Alog%283%2C%288%29%29 = log%283%2C%288%5E4%29%29 = log%283%2C%284096%29%29 > log%283%2C%282187%29%29 = log%283%2C%283%5E7%29%29 = 7.    Thus  log%283%2C%288%29%29 > 7%2F4.



From these two inequalities,  log%282%2C%283%29%29 < 7%2F4   and   log%283%2C%288%29%29 > 7%2F4,   you have   log%282%2C%283%29%29  <  log%283%2C%288%29%29.



Answer.  We have proven  log%282%2C%283%29%29  <  log%283%2C%288%29%29  without using calculator.

QED.