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Math Help - uniqueness of log

  1. #1
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    uniqueness of log

    Fix  b>1 and  y>0 . Prove that there is a unique real  x such that  b^{x} = y .

    So to show uniqueness we need to consider the following: if  b^{x} = y and  b^{l} = y then  y = l . But first we want to show that  b^{x} is monotonically increasing.

    So for any positive integer  n ,  b^{n}-1 \geq n(b-1) . We can show this by factoring the LHS as  (b-1)(b^{n-1}+b^{n-2}+b^{n-3} + \ldots + b^{2} + b+1) \geq (b-1)n . So  b-1 \leq \frac{b^{n}-1}{n} . Or  b-1 \geq n(b^{1/n}-1) . From here what do we do?
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  2. #2
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    In other words the strategy is that we need to consider a set....show that  x is the supremum of that set...and then show uniqueness?
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  3. #3
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    See what is the point of showing that is is strictly increasing? I see. Because you can show uniqueness by exploiting monotonicity.
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  4. #4
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    Let us consider the slightly different problem of showing that there is a unique  b . So we fix  x, y>0 . Then uniqueness follows because 0 < y_1 < y_2 \implies y_{1}^{n} < y_{2}^{n} . And to show existence we consider the set of  t 's such that  t^n < x .

    So we use this same set in our problem. But we are trying to use this strategy:  \text{monotonic increasing} \implies \text{uniqueness} .
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