SOLUTION: log base 2 x ^(log base 2 x) =4

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Question 71721: log base 2 x ^(log base 2 x) =4
Found 3 solutions by stanbon, MathTherapy, greenestamps:
Answer by stanbon(75887) About Me  (Show Source):
You can put this solution on YOUR website!
log(base 2) x ^(log(base 2) x) =4
--------
Converting to exponential you get:
x^(log(ba 2)x)=16
Taking the log you get:
(log(ba 2)x)(log)x = log16
[logx/log2](logx) = log 16
[logx]^2 = (log2)(log16)
[logx]^2 = 0.3624762332...
logx=0.6020599913...
x=10^0.6020599913...
x=4
===========
Cheers,
Stan H.

Answer by MathTherapy(10699) About Me  (Show Source):
You can put this solution on YOUR website!
log base 2 x ^(log base 2 x) =4

The other person only had 1 of the 2 solutions. Plus, the calculator was
used to get that answer. I don't think that that's what's expected!

log+%282%2C+%28x%5E%28log+%282%2C+%28x%29%29%29%29%29+=+4
       x%5E%28log+%282%2C+%28x%29%29%29+=+2%5E4 --------- Converting to EXPONENTIAL form
       x%5E%28log+%282%2C+%28x%29%29%29+=+16
          log+%282%2C+%28x%29%29+=+log+%28x%2C+%2816%29%29 ---- Converting to LOGARITHMIC form
          log+%282%2C+%28x%29%29+=+log+%28%2816%29%29%2Flog+%28%28x%29%29
          log+%282%2C+%28x%29%29+=+log+%282%2C+%2816%29%29%2Flog+%282%2C+%28x%29%29 ---- Converting right-side to base 2
          log+%282%2C+%28x%29%29+=+4%2Flog+%282%2C+%28x%29%29
      %28log+%282%2C+%28x%29%29%29%5E2+=+4 ------- Cross-multiplying
  sqrt%28%28log+%282%2C+%28x%29%29%29%5E2%29+=+0%2B-sqrt%284%29 ---- Taking sqrt of both sides
         log+%282%2C+%28x%29%29+=+0+%2B-+2
         log+%282%2C+%28x%29%29+=+2            OR    log+%282%2C+%28x%29%29+=+-+2
                 OR

Answer by greenestamps(13292) About Me  (Show Source):
You can put this solution on YOUR website!


Another path to the solution....

log%282%2C%28x%5Elog%282%2C%28x%29%29%29%29=4

Rewrite that using log%28%28A%5E%28%28B%29%29%29%29=B%2Alog%28%28A%29%29

log%282%2C%28x%29%29%2Alog%282%2C%28x%29%29=4
%28log%282%2C%28x%29%29%29%5E2=4

log%282%2C%28x%29%29=2 or log%282%2C%28x%29%29=-2

x=2%5E2=4 or x=2%5E%28-2%29=1%2F4

ANSWERS: x=4 or x=1/4