Question

Which of the following logarithmic functions are positive ?
i. ${\log }_{2}7$
ii. ${\log }_{0.2}3$
iii. ${\log }_{1/3}\left(1/5\right)$
iv. ${\log }_{4}3$
v. ${\log }_2\left({\log }_{2}9\right)$

Answer 1

i. Since, $7>4$

${\log }_{2}7>{\log }_2\left(4\right)$

${\log }_{2}7>2{\log }_{2}2$

${\log }_{2}7>2>0$

Hence, ${\log }_{2}7$ is positive.

ii. ${\log }_{0.2}3={\log }_{5^{-1}}\left(3\right)=-{\log }_{5}3<0$

iii. ${\log }_{1/3}\left(1/5\right)={\log }_{3^{-1}}{5}^{-1}={\log }_{3}5>0$

iv. Since, $3>1$

${\log }_{4}3>{\log }_4\left(1\right)=0$

$\Rightarrow {\log }_{4}3>0$

Hence, ${\log }_{4}3$ is positive.

v. Since, $9>8$

${\log }_{2}9>{\log }_{2}8$

${\log }_{2}9>{\log }_2{2}^3$

${\log }_{2}9>3$

${\log }_2\left({\log }_{2}9\right)>{\log }_{2}3>0$

Hence, ${\log }_2\left({\log }_{2}9\right)$ is positive.