Question

State whether each of the following statements is true or false.
(i) ${\log }_{5}125=3$
(ii) ${\log }_{\frac{1}{2}}8=3$
(iii) ${\log }_4(6+3)={\log }_{4}6+{\log }_{4}3$
(iv) ${\log }_2\left(\frac{25}{3}\right)=\frac{{\log }_{2}25}{{\log }_{2}3}$
(v) ${\log }_{\frac{1}{3}}3=-1$
(vi) ${\log }_a(M-N)={\log }_{a}M÷{\log }_{a}N$

Answer 1

$i){\log }_5{5}^3=3{\log }_{5}5=3\to True$

$ii){\log }_{\frac{1}{2}}8={\log }_{2^{-1}}{2}^3=\frac{3}{-1}{\log }_{2}2=-3\to False$

$iii\left){\log }_4\right(6+3)={\log }_{4}6+{\log }_{4}3\to False$

Because ${\log }_{4}6+{\log }_{4}3={\log }_4\left(6.3\right)$

$iv){\log }_2\left(\frac{25}{3}\right)=\frac{{\log }_{2}25}{{\log }_{2}3}\to False$

$v){\log }_{\frac{1}{3}}3={\log }_{3^{-1}}3=-1\to True$

$vi\left){\log }_a\right(M-N)={\log }_{a}M÷{\log }_{a}N\to False$