SOLUTION: Which result is correct? {{{ log( 3, ( log( 2, ( log( 2, 256 ) ) ) ) ) = x }}} {{{ log( 3, ( log( 2, ( log( 2, 2 ^ 8 ) ) ) ) ) = x }}} a) {{{ 3 ^ x = log( 2, ( log( 2, 2 ^ 8 ) )

Algebra ->  Logarithm Solvers, Trainers and Word Problems -> SOLUTION: Which result is correct? {{{ log( 3, ( log( 2, ( log( 2, 256 ) ) ) ) ) = x }}} {{{ log( 3, ( log( 2, ( log( 2, 2 ^ 8 ) ) ) ) ) = x }}} a) {{{ 3 ^ x = log( 2, ( log( 2, 2 ^ 8 ) )      Log On


   

Question 803391: Which result is correct?
+log%28+3%2C+%28+log%28+2%2C+%28+log%28+2%2C+256+%29+%29+%29+%29+%29+=+x+
+log%28+3%2C+%28+log%28+2%2C+%28+log%28+2%2C+2+%5E+8+%29+%29+%29+%29+%29+=+x+
a) +3+%5E+x+=+log%28+2%2C+%28+log%28+2%2C+2+%5E+8+%29+%29+%29+
+2+%5E+3+%5E+x+=+log%28+2%2C+2+%5E+8+%29+
+2+%5E+2+%5E+3+%5E+x+=+2+%5E+8+
+2+%5E+%28+6+%2A+x+%29+=+2+%5E+8+
+6+%2A+x+=+8+
+x+=+8+%2F+6+=+4+%2F+3+
b) +log%28+3%2C+%28+log%28+2%2C+8+%29+%29+%29+=x+
+log%28+3%2C+3+%29+=+x+
+x+=+1+
Where is the mistake?
Found 2 solutions by Edwin McCravy, KMST:
Answer by Edwin McCravy(20056) About Me  (Show Source):
You can put this solution on YOUR website!
Which result is correct? 
+log%28+3%2C+%28+log%28+2%2C+%28+log%28+2%2C+256+%29+%29+%29+%29+%29+=+x+
+log%28+3%2C+%28+log%28+2%2C+%28+log%28+2%2C+2+%5E+8+%29+%29+%29+%29+%29+=+x+

That is correct but not finished.  Here is the completion

+log%28+3%2C+%28+log%28+2%2C+%288+%29+%29+%29+%29++=+x+
+log%28+3%2C+%28+log%28+2%2C+%282%5E3+%29+%29+%29+%29++=+x+
+log%28+3%2C+%283+%29+%29+++=+x+
+log%28+3%2C+%283%5E1%29+%29++++=+x+
+1+=+x+

a) +3+%5E+x+=+log%28+2%2C+%28+log%28+2%2C+2+%5E+8+%29+%29+%29+
cross%28+2+%5E+3+%5E+x+=+log%28+2%2C+2+%5E+8+%29%29+

The preceding step is wrong. Exponentiation is not
associative.  Therefore we must always use parentheses 
in such cases since %28a%5Eb%29%5Ec and a%5E%28%28b%5Ec%29%29 are
not the same. To write a%5Eb%5Ec would be ambiguous, and
thus it is never used.  The corrected step is

+2+%5E+%28%283+%5E+x%29%29+=+log%28+2%2C+2+%5E+8+%29+

The completion is

+2+%5E+%28%283+%5E+x%29%29+=+8+

+2+%5E+%28%283+%5E+x%29%29+=+2%5E3+

Since the base 2 is positive and not equal to 1,
we may equate the exponents of 2:

3%5Ex+=+3

3%5Ex+=+3%5E1

and thus by the same rule:

x = 1


b) +log%28+3%2C+%28+log%28+2%2C+8+%29+%29+%29+=x+
+log%28+3%2C+3+%29+=+x+
+x+=+1+

This is correct.

Edwin

Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
I see no mistake in b), and it seems to be the most direct way to the answer.

In a), I believe that the problem is that
+3+%5E+x+=+log%28+2%2C+%28+log%28+2%2C+2+%5E+8+%29+%29+%29+ means that
+2+%5E+%28%283+%5E+x+%29%29=+log%28+2%2C+2+%5E+8+%29+
I see those parentheses around 3%5Ex as crucial to establish the proper meaning of the expression.

Certainly, +2+%5E+%28%283+%5E+x+%29%29 is not the same as %28+2+%5E+3+%29%5E+x+=+2%5E%286x%29 as you assumed to keep on going.
If x=5, +2+%5E+%28%283+%5E+x+%29%29=+2+%5E+%28%283+%5E5%29%29=2%5E243,
while %28+2+%5E+3+%29%5E+x+=+%28+2+%5E+3+%29%5E+5=8%5E5=2%5E15

Probably there are rules about order of operations that assign one or another meaning to 2%5E3%5Ex, but I would forget them soon after reading them, because those expressions do not come up too often, so I will use parentheses to make my meaning clear.

In any case, I would not know how to proceed further after getting to +2+%5E+%28%283+%5E+x+%29%29 , because I do not know of any equivalent expression.