While reading a script I found this equation:
$\gamma b^{e \log_b{n+e}} = \gamma b^e + n^e$
and i cannot figure out how the author did this. I'd appreciate a step-by step equation for this equation.
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2Looks wrong to me. Shouldn't you have a multiplication there instead of a sum on the right hand side? – Jyrki Lahtonen Feb 19 '13 at 18:02
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@Sim Is it $\log (n+e)$ or $\log (n)+e$? – Git Gud Feb 19 '13 at 18:08
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@GitGud no brackets used, but maybe due to convenience – Sim Feb 19 '13 at 18:10
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@JyrkiLahtonen could be, but not in the equation I got, it could be an error though – Sim Feb 19 '13 at 18:10
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@Sim ****ing analysts. Either way it seems wrong. It's "almost" right if it is $\log (n)+e$. – Git Gud Feb 19 '13 at 18:11
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@GitGud then treat it that way, like I said it could be $\log_b(n) + e and no brackets were used due to convenience (and for confusion) – Sim Feb 19 '13 at 18:15
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@Sim In that case Jyrki Lahtonen did that just above your comment. – Git Gud Feb 19 '13 at 18:15
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1@JyrkiLahtonen seems reasonable, I'd accept that as an answer, probably a typo or so – Sim Feb 19 '13 at 18:17
1 Answers
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By the definition of the logarithm $b^{\log_b n}=n$, so following the usual rules of powers of power: $b^{e\log_b n}=(b^{\log_b n})^e=n^e$. Another rule about powers is $b^{x+y}=b^x\cdot b^y$, so altogether we get $b^{e\log_b n+e}=b^{e\log_bn}\cdot b^e=n^e\cdot b^e$.
Therefore it should read $$\gamma b^{e\log_b n+e}=\gamma b^e\cdot n^e,$$ and it looks like a multiplication sign was accidentally converted to a plus sign.
Jyrki Lahtonen
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