SOLUTION: Is this solvable? log of 9 to the base of x equals log of 6 to the base of x

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Question 432348: Is this solvable?
log of 9 to the base of x equals log of 6 to the base of x
Found 2 solutions by richard1234, robertb:
Answer by richard1234(7193) About Me  (Show Source):
You can put this solution on YOUR website!
Yes, but no solution is expected:

log%28x%2C+9%29+=+log%28x%2C+6%29

Using the change of base formula,

log%289%29%2Flog%28x%29+=+log%286%29%2Flog%28x%29

log%289%29log%28x%29+=+log%286%29log%28x%29

log%28x%29%28log%289%29-log%286%29%29+=+0

log%28x%29+=+0 --> x+=+1. However, this would result in 9%2F0+=+6%2F0, not a very nice equation since z%2F0 can be infinite, hence undefined. Therefore there are no real solutions.

Answer by robertb(5830) About Me  (Show Source):
You can put this solution on YOUR website!
Assuming x+%3E+0 and x%3C%3E1, (the parameters for the logarithmic base), the expression log%28x%2C+b%29 is one-to-one, and so by property of inverses,
log%28x%2C9%29+=+log%28x%2C+6%29 <==> x%5Elog%28x%2C9%29+=+x%5Elog%28x%2C6%29, or 9=6, which is false. Hence the equation has no real solution.