Question 221046: how do u do 4^(log(base 8) of 27 without a calculator
Problem: 4^(log(27)/log(8))
Found 2 solutions by jim_thompson5910, Edwin McCravy:
Answer by jim_thompson5910(35256)   (Show Source):
Answer by Edwin McCravy(20060)  (Show Source):
You can put this solution on YOUR website! Edwin's full explanation:
Since there is a  and an  involved, and
both are powers of  , let's try to get as much
as possible involving the number :
Let's first work only on the exponent
Use the change of base formula 
to change base to base
Write  as  and as 
Use the rule  to rewrite
numerator and denominator:
Cancel the  's:
and we have just:
Use the rule  to rewrite the denominator:
or just
Now let's go back to the original problem, which was:
and replace the exponent with
Rewrite the base as 
Use the rule 
Use the rule  to rewrite the exponent:
Write  as 
Use the rule 
Final answer:
--------------------------------
Again we notice there is a and an involved,
and both are powers of , let's try to get as much
as possible involving the number :
As before we first work only with the exponent:
We write as and as
Use the rule to rewrite
numerator and denominator:
Cancel the  's
When no base is written, the base is assumed to be 
Write that fraction as a division:
  
Use the change of base formula: on each to get them
to logs base 2.
 
Use the rule  to rewrite 
Invert the second fraction and change division to multiplication:
 
Cancel:
 
All that's left is
Now let's go back to the original problem, which was:
and replace the exponent by the simpler
Wow! That just happens by chance to be the exact same
thing as we got in the first problem, so we just copy
the above from there on and, of course, it will have
the same final answer.
Rewrite the base as
Use the rule
Use the rule to rewrite the exponent:
Write as
Use the rule
Final answer:
Edwin
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