SOLUTION: how do u do 4^(log(base 8) of 27 without a calculator Problem: 4^(log(27)/log(8))

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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) About Me  (Show Source):
You can put this solution on YOUR website!
In order to evaluate , we need to simplify the exponent


... Start with the given expression.


... Use the change of base formula.


... Rewrite 27 as 3%5E3 and rewrite 8 as 2%5E3


... Pull down the exponents using the identity log%28b%2C%28x%5Ey%29%29=y%2Alog%28b%2C%28x%29%29


... Cancel out the common terms.


... Use the change of base formula.


So


----------------------------------------------------------------------


... Start with the partially simplified expression.


... Rewrite 4 as 2%5E2


... Multiply the exponents.


... Place the coefficient as the exponent using the identity y%2Alog%28b%2C%28x%29%29=log%28b%2C%28x%5Ey%29%29


... Use the identity to simplify.


... Square 3 to get 9.


So


===================================

Answer:


This means that

Answer by Edwin McCravy(20060) About Me  (Show Source):
You can put this solution on YOUR website!
Edwin's full explanation:


4%5E%28log%288%2C27%29%29

Since there is a 4 and an 8 involved, and
both are powers of 2, let's try to get as much
as possible involving the number 2:

Let's first work only on the exponent

log%288%2C27%29

Use the change of base formula 

to change base 8 to base 2

log%282%2C27%29%2Flog%282%2C8%29

Write 27 as 3%5E3 and 8 as 2%5E3

log%282%2C3%5E3%29%2Flog%282%2C2%5E3%29

Use the rule log%28%28B%29%2CX%5EN%29=N%2Alog%28%28B%29%2CX%29 to rewrite
numerator and denominator:

%283%2Alog%282%2C3%29%29%2F%283%2Alog%282%2C2%29%29

Cancel the 3's:

%28cross%283%29%2Alog%282%2C3%29%29%2F%28cross%283%29%2Alog%282%2C2%29%29

and we have just:

%28log%282%2C3%29%29%2F%28log%282%2C2%29%29

Use the rule log%28B%2C%28B%29%29=1 to rewrite the denominator:

%28log%282%2C3%29%29%2F1

or just

log%282%2C3%29

Now let's go back to the original problem, which was:

4%5E%28log%288%2C27%29%29

and replace the exponent with log%282%2C3%29

4%5Elog%282%2C3%29

Rewrite the base 4 as %282%5E2%29

%282%5E2%29%5Elog%282%2C3%29

Use the rule %28B%5EM%29%5EN=B%5E%28M%2AN%29

2%5E%282%2Alog%282%2C3%29%29

Use the rule N%2Alog%28%28B%29%2CX%29=log%28%28B%29%2CX%5EN%29 to rewrite the exponent:

2%5E%28log%282%2C3%5E2%29%29

Write 3%5E2 as 9

2%5E%28log%282%2C9%29%29

Use the rule log%28B%2C%28B%5EN%29%29+=+N

Final answer: 

9

--------------------------------



Again we notice there is a 4 and an 8 involved,
and both are powers of 2, let's try to get as much
as possible involving the number 2:

As before we first work only with the exponent:

log%2827%29%2Flog%288%29

We write 27 as 3%5E3 and 8 as 2%5E3

log%283%5E3%29%2Flog%282%5E3%29

Use the rule log%28%28B%29%2CX%5EN%29=N%2Alog%28%28B%29%2CX%29 to rewrite
numerator and denominator:

%283%2Alog%283%29%29%2F%283%2Alog%282%29%29

Cancel the s's

%28cross%283%29%2Alog%283%29%29%2F%28cross%283%29%2Alog%282%29%29

log%283%29%2Flog%282%29

When no base is written, the base is assumed to be 10

log%2810%2C3%29%2Flog%2810%2C2%29

Write that fraction as a division:

log%2810%2C3%29%22%F7%22log%2810%2C2%29

Use the change of base formula:  on each to get them
to logs base 2.

log%282%2C3%29%2Flog%282%2C10%29%22%F7%22log%282%2C2%29%2Flog%282%2C10%29

Use the rule log%28B%2C%28B%29%29+=+1 to rewrite log%282%2C2%29

log%282%2C3%29%2Flog%282%2C10%29%22%F7%221%2Flog%282%2C10%29

Invert the second fraction and change division to multiplication:

log%282%2C3%29%2Flog%282%2C10%29%22%D7%22log%282%2C10%29%2F1

Cancel:

log%282%2C3%29%2Fcross%28log%282%2C10%29%29%22%D7%22cross%28log%282%2C10%29%29%2F1

All that's left is 

log%282%2C3%29

Now let's go back to the original problem, which was:



and replace the exponent by the simpler log%282%2C3%29

4%5Elog%282%2C3%29 

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 4 as %282%5E2%29

%282%5E2%29%5Elog%282%2C3%29

Use the rule %28B%5EM%29%5EN=B%5E%28M%2AN%29

2%5E%282%2Alog%282%2C3%29%29

Use the rule N%2Alog%28%28B%29%2CX%29=log%28%28B%29%2CX%5EN%29 to rewrite the exponent:

2%5E%28log%282%2C3%5E2%29%29

Write 3%5E2 as 9

2%5E%28log%282%2C9%29%29

Use the rule log%28B%2C%28B%5EN%29%29+=+N

Final answer: 

9 

Edwin