???? ?? ???? ????? ??????? ??????? ????? ????????. ?????? ??????? ??? ????? ????? .

??????? (Logarithm) ??? ??????? ????? ? ??????? ???????? . ???????? ?????? ??????? ${\displaystyle \log }$. ????? ?? ????? ?? ? ????? ?????? ???? ??????, ????? ??? ???? ?? ?????: "????? ???? ?? ?????? ?? ????? b ??? ???? ?? ?????? a?". ??????? ?? a ??? ???? b ???? ${\displaystyle \log _{b}a}$.

????? [ ????? ??? ???? | ????? ]

?????, ???????? ?? ${\displaystyle x}$ ? ???? ???? ${\displaystyle \ b}$ (????? ???? ????? ?-1 ?????? x ?????), ??? ? ???? ??? ?? ?????? ?? ${\displaystyle \ b}$ ??? ???? ?? ${\displaystyle \ x}$.
?? ${\displaystyle \ b>0}$ ??? ${\displaystyle \ b^{y}=x}$, ???? ?? ${\displaystyle \ y}$ ??? ??????? ?? ${\displaystyle \ x}$ ??? ???? ${\displaystyle \ b}$, ?????? ???: ${\displaystyle \ y=\log _{b}x}$.

??, ????? ?????????? ??????? ???: ${\displaystyle \ b^{log_{b}x}=x}$.

?????: ${\displaystyle \ \log _{10}100=2}$, ???? ?- ${\displaystyle \ 100=10^{2}}$.

???????: ???????? ?? ${\displaystyle \ 100}$ ??? ???? ${\displaystyle \ 10}$ ??? ${\displaystyle \ 2}$, ???? ?? ${\displaystyle \ 10}$ ????? ${\displaystyle \ 2}$ ???? ?? ${\displaystyle \ 100}$.

???? ????? ?????:

????? ?????? ${\displaystyle \ \log }$ ??? ???? ????? ???? ????? ???, ???? ???????? ????? ??? ?????:

• ??????? ??????? ??????? ${\displaystyle \ \log _{e}}$ (? ??????? ????? , ?????? ???? ????? ). ???? ????? ??????? ?? ?????? ${\displaystyle \ \ln }$.
• ??????? ??????? ??????? ${\displaystyle \ \log _{10}}$, ?????? ???? ?????? ${\displaystyle \ {\mbox{lg}}}$.
• ??????? ??????? ? ???? ????? ??????? ${\displaystyle \ \log _{2}}$, ?????? ???? ?????? ${\displaystyle \ {\mbox{lb}}}$.

??? ??????? [ ????? ??? ???? | ????? ]

?????????? ?????? ?????? ?????? ???? ?-17 ?? ??? ?'?? ????? ?????? . ?? ??????? ?? ??? ? ???? ?? ?????? , ???? ???? ?? ???? ?-20 , ??? ?????????? ??? ??? ????? ? ????? , ??????? ??? ????????? ???????? ???? ????? . ?????? ?????? ?????? ??? ???? ????? ??? ??? ???? ???? ??????? ?? ????? ???? ????? ?????????? ?? ?? ??? ?????? ??????. ??? ?? ????? ?????? ????? ??? , ???? ????? ?????? ?????, ?????? ?????? ?????? ????. ????? ??? ???? ????? ?????????? ??? ???? ${\displaystyle \ 10}$.

???? ?????????? [ ????? ??? ???? | ????? ]

?????? ???????? ???? ?????? ??? ${\displaystyle \ a,b,c,d}$ ?????? ???????, ?????? ????? ?????????? ???? ??1.

????? ???????	${\displaystyle \ \log _{a}(1)=0}$ ${\displaystyle \ \log _{a}(a)=1}$
???, ????? ?????? ????? ????? ??? ?????? ???? ?? ?????? ???, ?????, ???? ?????, ??????? ??? ????????? ?? ???? ?????.	${\displaystyle \ \log _{c}(a\cdot b)=\log _{c}(a)+\log _{c}(b)}$ ${\displaystyle \ \log _{c}\left({\frac {a}{b}}\right)=\log _{c}(a)-\log _{c}(b)}$ ${\displaystyle \ {\log _{a}b}\cdot {\log _{c}d}={\log _{c}b}\cdot {\log _{a}d}}$ ??? ${\displaystyle \ r}$ ???? : ${\displaystyle \ \log _{c}(a^{r})=r\cdot \log _{c}(a)}$
???????? ????????? ???????? ????? ??? ???????? ?????? ??????? ??? ????? ??? "?????" ?? ????.	${\displaystyle \ x^{\log _{a}b}=b^{\log _{a}x}}$ ${\displaystyle \ a^{\log _{a}b}=b}$ ??? ${\displaystyle \ r}$ ???? : ${\displaystyle \ \log _{a}a^{r}=r}$
????? ???? ???????? ??? ?? ?????? ????? ????????? ?????????. ???? ????????? ?? ?????? ?????? ??????? ???? (ln) ???????? ????? 10 ( ${\displaystyle \ \log _{10}}$), ??? ?? ????? 2 ( ${\displaystyle \ \log _{2}}$). ??? ???? ?? ${\displaystyle \ \log _{2}100}$, ??????: ${\displaystyle \ {\frac {\log _{10}100}{\log _{10}2}}}$ ?? ${\displaystyle \ {\frac {\ln {100}}{\ln {2}}}}$ ????? ????? ???.	${\displaystyle \ \log _{a}b={\frac {\log _{c}b}{\log _{c}a}}}$ ${\displaystyle \ \log _{a}b={\frac {1}{\log _{b}a}}}$
??????	???? a > 1: ${\displaystyle \ \lim _{x\to 0}\log _{a}(x)=-\infty }$ ???? a < 1: ${\displaystyle \ \lim _{x\to 0}\log _{a}(x)=\infty }$ ???? a > 1: ${\displaystyle \ \lim _{x\to \infty }\log _{a}(x)=\infty }$ ???? a < 1: ${\displaystyle \ \lim _{x\to \infty }\log _{a}(x)=-\infty }$ ${\displaystyle \ \lim _{x\to 0}\log _{a}(x)\cdot x^{b}=0}$ ${\displaystyle \ \lim _{x\to \infty }\log _{a}(x)/x^{b}=0}$
?????	${\displaystyle \ {d \over dx}\log _{a}(x)={\frac {1}{\ln(a)}}{1 \over x}}$
???????	${\displaystyle \ \int \log _{a}(x)\,dx=x\log _{a}(x)-{\frac {x}{\ln(a)}}+C}$

????? ?????? ????????? [ ????? ??? ???? | ????? ]

????????? ?? ?????? [ ????? ??? ???? | ????? ]

???'?? ?????? ????? ???????? ?????? ???? ????????? ????? ????? ??????? ????? ?????? ????? ???? ?????? ???? ?????? ????? ?????????? ??????, ???? ?????? ???? ??????? ????? ????????. ?????? ?? ????????? ????? ?? ?????? ??? ???? ???? ????? ??? 1 ?-2 ???? ?????? ??????? ?????? ${\displaystyle (1+2^{-k})}$ ????? ????. ????????? ???? ?????? ${\displaystyle P}$ ????? ?????: ?? ${\displaystyle P\cdot (1+2^{-k})<x}$, ?? ????? ?? P ?- ${\displaystyle P^{*}=P\cdot (1+2^{-k})}$. ???? ??? ?????? ?? ????? ??? ?????? ?? ????? ????? ?????? ??????? ??? ?????? ${\displaystyle (1+2^{-(k+1)})}$. ?? ${\displaystyle P^{*}\cdot (1+2^{-(k+1)})<x}$ ?? ?????? ??? ???? ?????? ?? x ???????. ????????? ????? ???? k ???? ????? ??? ????? ?? ????? ?????. ????? ?- ${\displaystyle \log(x)}$ ??? ???? ?? ?????? ?????? ${\displaystyle \log(1+2^{-k})}$ ??? ?????? ?????? ??????? ?? x, ?? ${\displaystyle \log(x)}$ ???? ?????? ????? ????? ????, ????? ???? ????? ?? ${\displaystyle \log(1+2^{-k})}$ ???? ?? ???? k.

???? : ????????? ???? ????? ?????? ????????? ??????? (???? ????? ?????????), ??? ???? ???? ???? ???? ?? ?????? ??????????? ????????, ?? ??? ???? "????" ?? ????? ??????? "??????" ????.

?????? ????????-??????? [ ????? ??? ???? | ????? ]

?????? ????????-??????? ???? ??????? ?????? ?????? ???? ???????? ?????. ??????? Sasaki ?-Kanada ???? ?-1982 ???????? ???????? ?? ?????? ???????? ??????? ?????? ?????? ???? ???? ??? 400 ?-1000 ????? ????????, ???? ?????? ???????? ?? ??? ?????? ?? ?????? ???? ???? ???? ???? ????. ??????? (ln(x ????? ????? ?? ${\displaystyle 2^{-p}}$ ????? ?????? ???? (????? ? ??? ??????? ???? ):

${\displaystyle \ln(x)\approx {\frac {\pi }{2M(1,2^{2-m}/x)}}-m\ln(2)}$

???? m ???? ?? ?- ${\displaystyle x\,2^{m}>2^{p/2}\,}$ ??? ?????? ?? ????? ?????.

??? ?? [ ????? ??? ???? | ????? ]

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• ??? ?????, ??????? - ??????, ?????? ??????? (???? PDF ?? ??? ????????)
• ??? ???????????', ?? ?? ????????? ?????? ?? ?? ???? , ???? "?? ?????", 8 ????? 2020
• How I Feel About Logarithms ? ????? ???? ?? Vi Hart ?? ????????? (???????)
• ??????? , ???? MathWorld (???????)
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