Démonstration des propriétés du logarithme

$\large\log_b(b^c)=c$log, start base, b, end base, left parenthesis, b, start superscript, c, end superscript, right parenthesis, equals, c

$\begin{aligned}\log_2({4\times 8})&=\log_2(2^2\times 2^3)&&\small{\gray{2^2=4\text{ et } 2^3=8}}\\ \\ &=\log_2(2^{2+3})&&\small{\gray{\text{$a^m\times a^n=a^{m+n}$}}}\\ \\ &=2+3&&\small{\gray{\text{$\log_b(b^c)=c$}}}\\ \\ &=\log_2(4)+\log_2(8)&&\small{\gray{\text{car $2=\log_2(4)$ et $3=\log_2(8)$}}}\\ \end{aligned}$

$\begin{aligned}\log_b(MN)&=\log_b(b^x\times b^y)&&\small{\gray{\text{}}}\\ \\ &=\log_b(b^{x+y})&&\small{\gray{\text{}}}\\ \\ &=x+y&&\small{\gray{\text{car $\log_b(b^c)=c$}}} \\\\ &=\log_b(M)+\log_b(N)&&\small{\gray{\text{}}} \end{aligned}$

$\begin{aligned}\log_b\left(\dfrac{M}{N}\right)&=\log_b\left(\dfrac{b^x}{ b^y}\right)&&\small{\gray{\text{}}}\\ \\ &=\log_b(b^{x-y})&&\small{\gray{\text{}}}\\ \\ &=x-y&&\small{\gray{\text{car $\log_b(b^c)=c$}}}\\ \\ &=\log_b(M)-\log_b(N)&&\small{\gray{\text{}}} \end{aligned}$

$\begin{aligned}\log_b\left(M^p\right)&=\log_b(\left({b^x}\right)^p)&&\small{\gray{\text{}}}\\ \\ &=\log_b(b^{xp})&&\small{\gray{\text{}}}\\ \\ &=xp&&\small{\gray{\text{$\log_b(b^c)=c$}}}\\ \\ &=\log_b(M)\times p&&\small{\gray{\text{}}}\\ \\ &=p\times\log_b(M)&&\small{\gray{\text{}}} \end{aligned}$