In the solution to exercise 1.2.2, 10 in Knuth’s The Art of Computer Programming, he states
If $log_{10}2 = \frac{p}{q}$, with $p$ and $q$ positive, then $2^q$ = $10^p$ (...)
Where does this relationship come from?
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Philipp Ludwig
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2Do you know how to rewrite $\log_{10}2=x$ as an exponential equation? Do you know what $\log_{10}y=z$ means? – Gerry Myerson Mar 22 '22 at 09:05
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1$log_{10}2 = \frac{p}{q} \iff 2=10^{\frac{p}{q}}$ – Fred Mar 22 '22 at 09:05
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Apply the base 10 exponential to both sides of the equality and then raise to the power $q$. – PierreCarre Mar 22 '22 at 09:06
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@GerryMyerson I know that i could rewrite it as $2 = 10^x$, but that does not help me to understand this. – Philipp Ludwig Mar 22 '22 at 09:07
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1Sure it does. What if you do the same thing with $\log_{10}2=p/q$? – Gerry Myerson Mar 22 '22 at 09:10
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1@PierreCarre and Gerry Myerson Thank you to both of you, now I get it. – Philipp Ludwig Mar 22 '22 at 09:12
1 Answers
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With the help of PrierreCarre and Gerry Myerson:
$ log_{10}2 = \frac{p}{q} \\ \Leftrightarrow 2 = 10^\frac{p}{q} \\ \Leftrightarrow 2^q = 10^p $
Philipp Ludwig
- 150