SOLUTION: logbase4logx(base2) + logbase2logx(base4) = 2. Find x.

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Question 1135088: logbase4logx(base2) + logbase2logx(base4) = 2. Find x.
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

log%284%2C%28log%282%2Cx%29%29%29+%2B+log%282%2C%28log%284%2Cx%29%29%29+=+2

Apply log rule : log+%28a%2Cb+%29=+log%28c%2Cb+%29%2Flog+%28c%2Ca+%29

log%282%2C%28log%284%2Cx%29%29%29=log%284%2C%28log%284%2Cx%29%29%29%2Flog%284%2C2%29
log%284%2C%28log%282%2Cx%29%29%29+%2Blog%284%2C%28log%284%2Cx%29%29%29%2Flog%284%2C2%29=2....simplify
log%284%2C%28log%284%2Cx%29%29%29%2Flog%284%2C2%29=2log%284%2C%28log%284%2Cx%29%29%29

so, you have

log%284%2C%28log%282%2Cx%29%29%29+%2B2log%284%2C%28log%284%2Cx%29%29%29=2
log%284%2C%28log%282%2Cx%29%29%29+%2Blog%284%2C%28log%284%2Cx%29%29%5E2%29=2


Apply log rule : log%28c%2Ca+%29%2Blog%28c%2Cb+%29=log%28c%2C%28ab%29%29
so
log%284%2C%28log%282%2Cx%29%29%2A%28log%284%2Cx%29%29%5E2%29=2
Apply log rule : a=log%28b%2Cb%5Ea%29 and write 2=log%284%2C4%5E2%29=log%284%2C16+%29


log%284%2C%28log%282%2Cx%29%29%2A%28log%284%2Cx%29%29%5E2%29=log%284%2C16+%29
When the logs have the same base: log+%28b%2Cf%28x%29%29=log%28b%2Cg%28x%29%29=>f%28x%29=g%28x%29
solog%282%2Cx%29%2A%28log%284%2Cx%29%29%5E2=16+

write log+%282%2Cx%29as log+%284%2Cx%29%2Flog%284%2C2%29

%28log+%284%2Cx%29%2Flog%284%2C2%29%29%2A%28log%284%2Cx%29%29%5E2=16+

%28log+%284%2Cx%29%29%5E3%2Flog%284%2C2%29=16+

%28log+%284%2Cx%29%29%5E3%2F%281%2F2%29=16+

2%28log+%284%2Cx%29%29%5E3=16+
%28log+%284%2Cx%29%29%5E3=8+

For %28g+%28x+%29+%29%5En=f+%28a+%29 whenn is odd, the solution is :g%28x%29=root%28n%2Cf%28a%29%29

log+%284%2Cx%29=root%283%2C8%29+
log+%284%2Cx%29=2+....change to base 10

log+%28x%29%2Flog+%284%29=2+
log+%28x%29=2log+%284%29+
log+%28x%29=log+%284%5E2%29+
log+%28x%29=log+%2816%29+...if log same,=>
x=16