Question 82835
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prove the power property: log<sub>b</sub>U<sup>n</sup> = n·log<sub>b</sub>U

Proof is by induction:

First we prove it's true for n = 1.

log<sub>b</sub>U<sup>1</sup> = log<sub>b</sub>U = 1·log<sub>b</sub>U

Assume true for n <u><</u> k,

Need to prove that log<sub>b</sub>U<sup>k+1</sup> = (k+1)log<sub>b</sub>U

log<sub>b</sub>U<sup>k+1</sup> = log<sub>b</sub>(U<sup>k</sup>U<sup>1</sup>) = log<sub>b</sub>U<sup>k</sup> + log<sub>b</sub>U<sup>1</sup>

= k·log<sub>b</sub>U + log<sub>b</sub>U =  (log<sub>b</sub>U)(k+1) = (k+1)log<sub>b</sub>U

Edwin</pre>