Question 955354
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Which number is greater:  {{{log(2,(3))}}}  or  {{{log(3,(8))}}} ?
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<pre>
We have, from one side,

{{{4*log(2,(3))}}} = {{{log(2,(3^4))}}} = {{{log(2,(81))}}} < {{{log(2,(128))}}} = 7.   Thus  {{{log(2,(3))}}} < {{{7/4}}}.



We have, from the other side,

{{{4*log(3,(8))}}} = {{{log(3,(8^4))}}} = {{{log(3,(4096))}}} > {{{log(3,(2187))}}} = {{{log(3,(3^7))}}} = 7.    Thus  {{{log(3,(8))}}} > {{{7/4}}}.



From these two inequalities,  {{{log(2,(3))}}} < {{{7/4}}}   and   {{{log(3,(8))}}} > {{{7/4}}},   you have   {{{log(2,(3))}}}  <  {{{log(3,(8))}}}.



<U>Answer</U>.  We have proven  {{{log(2,(3))}}}  <  {{{log(3,(8))}}}  without using calculator.
</pre>

QED.