Tag: physics

$\log _{ 10 }{ 0.01= } \log _{ 10 }{ { 10 }^{ -2 } } $

$ \log _{ 10 }{ 1 }= 0\ 1 = { 10 }^{ 0 }$

$ \log _{ 5 }{ 125 } =x\ 125 = { 5 }^{ x }\ x = 3 $

$ \log _{ 5 }{ 1 } =x\ 1\quad =\quad { 5 }^{ x }\ x\quad =\quad 0\ \ $

$ \log _{ 10 }{ 0.01 } = -2 \Rightarrow 0.01 = { 10 }^{ -2 } $

$y={ a }^{ x }\Rightarrow \log _{ a }{ y } =x\ \therefore 81={ 3 }^{ 4 }\Rightarrow \log _{ 3 }{ 81 } =4$

$log 27 = 1.431$
$\Rightarrow log (3^3) = 1.431$
$\Rightarrow 3 log 3 = 1.431$
$\Rightarrow log 3 = 0.477$
$\therefore log 9 = log (3^2) = 2 log 3 = (2 \times 0.477) = 0.954$

Given, $\log _c a=x$

(a) Since $log _a a = 1,$ so $log _{10}  10 = 1$
(b) $log (2 + 3) log 5$ and $log (2 \times 3) = log 6 = log 2 + log 3$
$\therefore log(2 + 3) \neq log (2 \times 3)$
(c) Since, $log _a  1 = 0$, so, $log _{10} 1 = 0$.
(d) $log(1 + 2 + 3) = log 6 = log (1 \times 2 \times 3) = log 1 + log 2 + log 3$

$log _{10} 80 = log _{10} (8 \times 10)$
$= log _{10} 8 + log _{10} 10$
$=log _{10} (2^3) + 1$
$= 3 log _{10} 2 + 1$
$= (3 \times 0.3010) + 1$
$= 1.9030$