Fundamental Laws of Logarithms

Q. If ${\log }_{2}x+{\log }_{8}x+{\log }_{64}x=3$, then the value of $x$ is

1. $1$
2. $2$
3. $3$
4. $4$

Q. The value of ${\log }_{\sqrt{2}}\left({\log }_{\sqrt{2}}\right({\log }_{\sqrt[4]{43}}\left({\log }_{3}27\right)\left)\right)$ is

1. $2$
2. $3$
3. $4$
4. $\sqrt[3]{33}$

Q. The value of ${\left(\right({\log }_{2}9{)}^2)}^{\frac{1}{{\log }_2\left({\log }_{2}9\right)}}\cdot {\left(\sqrt{7}\right)}^{\frac{1}{{\log }_{4}7}}$ is

1. $\sqrt{7}{\log }_{2}9$
2. $8$
3. ${\log }_{9}7$
4. $63$

Q. The value of $x$, if ${\log }_{\sqrt{2}}\left({\log }_2\right({\log }_4(x-15)\left)\right)=0$

1. $16$
2. $31$
3. $19$
4. $17$

Q. The value of the expression ${\log }_{\pi }{\pi }^7+{\log }_{1/\pi }{\pi }^3$ is

Q. The solution to the equation ${\log }_{10}\sqrt{x-1}+\frac{1}{2}{\log }_{10}(2x+15)=1$ is

1. $-\frac{3}{2}$
2. $-\frac{23}{2}$
3. $5$
4. both $5$ and $-\frac{23}{2}$

Q. If $(5.6{)}^x=(0.0056{)}^y={10}^5$, then the value of $\frac{1}{x}-\frac{1}{y}$ is

1. $\frac{1}{5}$
2. $\frac{3}{5}$
3. $\frac{5}{6}$
4. $\frac{5}{3}$

Q. If $\frac{\ln x}{b-c}=\frac{\ln y}{c-a}=\frac{\ln z}{a-b}$, then which of the following option(s) is/are true ?

1. $xyz=1$
2. $x^a{y}^b{z}^c=1$
3. $x^{b+c}{y}^{c+a}{z}^{a+b}=1$
4. $xyz=x^a{y}^b{z}^c$

Q. The value of $lo{g}_{b}a.lo{g}_{c}b.lo{g}_a$ c is

1. 0
2. log abc
3. 1
4. 10

Q. The value of the expression ${\log }_{8}512+\left[\frac{\log 7+\log 42-\log 6}{\log 49}\right]$ is