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34.2 Math Problems Out of Nowhere
, which we used to prove our answer to the previous problem. Show that
. [Recall: log means an unspecified base]
We have that
- Taking the log of both sides, we have that.
- Bringing down the exponent we have that.
- Discarding the unnecessary constant, we have.
- From there, we have that. [sinceis the same thing asymptotically as]
In other words,
grows at least as quickly as
We’ve shown that
- In other words, these two functions grow at the same rate asymptotically.
As for why we did this, we will see in a little while...