Contents
<aside> <img src="/icons/book_orange.svg" alt="/icons/book_orange.svg" width="40px" /> A compendium for Mathematics NS1 written by Martin Deraas has been published on the Amazon store. It is available for purchase as a paperback for £9.95 (120 NOK), and as a PDF document on Google Drive/Adobe Acrobat for NOK 100.
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<aside> <img src="https://image.flaticon.com/icons/svg/1828/1828885.svg" alt="https://image.flaticon.com/icons/svg/1828/1828885.svg" width="40px" /> For first 1,000 digits of e, see ‣.
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Mathematical expression of a logarithm:
$$ n^{\log_n a}=a $$
From this we can conclude that:
$$ \begin{alignedat}{1} {\log_n a}=b&\Leftrightarrow a=n^b\\ a<b&\Leftrightarrow \log_n a<\log_n b \end{alignedat} $$
Exponents
$$ \begin{alignedat}{1} \log_n\space a^x&=x\space\log_n\space a\\ a^x=b&\Leftrightarrow x={\log_n\space b\over \log_n\space a} \end{alignedat} $$
Multiplication and Division
$$ \begin{alignedat}{3} \log_n (a\cdot b)&\space =\space &\log_n a&\space +\space &\log_n b\\[5pt] \log_n\lbrack{a\over b}\rbrack &\space =\space &\log_n a&\space -\space &\log_n b \end{alignedat} $$
Common
Also known as the Briggsian logarithm, named after its inventor Henry Briggs.
Base: 10
$$ \lg a=\log_{10} a $$
Natural
Base: e, that of the natural logarithm;
$$ \ln a=\log_e a $$
Definition:
$$ e=\lim_{t\to0}(1+t)^{1\over t}≈2.71828 $$
Derivative of ln(e^x):
$$ {d\over dx}\lbrack \ln (e^x)\rbrack ={1\over e^x}\cdot (e^x)' $$
Derivative of (e^x):
$$ 1={(e^x)'\over e^x},(e^x)'=e^x $$
$$ \int e^x\space dx = e^x+C $$
