Fundamental Laws of Logarithms

Q. If $\log \frac{a}{b}+\log \frac{b}{a}=\log (a+b),$ then

1. $a+b=0$
2. $a+b=1$
3. $a-b=0$
4. $a-b=1$

Q. The solution of the equation $lo{g}_{7}lo{g}_5\left(\sqrt{x^2+5+x}\right)=0$

1. x = 2
2. x = 3
3. x = 4
4. x =-2

Q. The value of $6+{\log }_{\frac{3}{2}}\left(\frac{1}{3\sqrt{2}}\sqrt{4-\frac{1}{3\sqrt{2}}\sqrt{4-\frac{1}{3\sqrt{2}}\sqrt{4-\frac{1}{3\sqrt{2}}\dots \dots }}}\right)$ is

1. $6$
2. $8$
3. $2$
4. $4$

Q. The value of $5^{{\log }_{\frac{1}{5}}\left(\frac{1}{2}\right)}+{\log }_{\sqrt{2}}\left(\frac{4}{\sqrt{3}+\sqrt{7}}\right)+{\log }_{\frac{1}{2}}\left(\frac{1}{10+2\sqrt{21}}\right)$ is

1. $3$
2. $4$
3. $5$
4. $6$

Q.

Logarithm of $32\sqrt[5]{4}$ to the base $2\sqrt{2}$ is

1. $3.6$

2. $5$

3. $4$

4. None of these

Q. If ${\log }_x\left({\log }_4\right({\log }_x(5x^2+4x^3)\left)\right)=0$, then the value of $x$ is

1. $2$
2. $4$
3. $3$
4. $5$

Q. If $lo{g}_{7}2=m,thenlo{g}_{49}28$ is equal to

1. $2(1+2m)$
2. $\frac{1+2m}{2}$
3. $\frac{2}{1+2m}$
4. $1+m$

Q. The equation $4{\log }_{\frac{x}{2}}\left(\sqrt{x}\right)+2{\log }_{4x}\left(x^2\right)=3{\log }_{2x}\left(x^3\right)$ is having

1. all rational roots
2. two rational and one irrational roots
3. one rational and two irrational roots
4. all irrational roots

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 ${\log }_5{\log }_3\sqrt{\sqrt[5]{59}}$ is

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

Q. If $x^{{\log }_3{x}^2+({\log }_{3}x{)}^2-10}=\frac{1}{x^2}$, then number of value(s) of $x$ satisfying the equation is/are

Q. The equation $\left(\sqrt{1+{\log }_x\sqrt{27}}\right){\log }_{3}x+1=0$ has

1. no integral solution
2. one irrational solution
3. two real solutions
4. no prime solution

Q. If ${\log }_{4}A={\log }_{6}B={\log }_9(A+B)$, then $\left[\frac{4B}{A}\right]$ (where $[.]$ represents the greatest integer function) equals

Q. If $A=\{x:x^{{\log }_{\sqrt{x}}2x}=4\}$, then $n\left(A\right)=$

1. $1$
2. $2$
3. $0$
4. infinitely many

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. If $3+{\log }_{5}x=2{\log }_{25}y$, then the value of $x$ is

1. $\frac{y}{25}$
2. $\frac{2y}{125}$
3. $\frac{y}{75}$
4. $\frac{y}{125}$

Q. The number of values of $x$ satisfying $1+{\log }_5(x^2-9)={\log }_5(x^2+4x+3)$ is

1. $0$
2. $1$
3. $2$
4. infinitely many

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. If ${\log }_{10}7=0.8451$, then the position of the first significant figure of $7^{-20}$, is

1. $15$
2. $20$
3. $17$
4. $18$

Q. The number of real values of $x$, that satisfy the equation $x^{{\log }_{\sqrt{x}}2x}=4$ is

1. $0$
2. $1$
3. $2$
4. $3$

Q. Let $(x_1,y_1,z_1)$ and $(x_2,y_2,z_2)$ be 2 sets of solution satisfying the following equations:
${\log }_{10}\left(2xy\right)=4+({\log }_{10}x-1)({\log }_{10}y-2)$
${\log }_{10}\left(2yz\right)=4+({\log }_{10}y-2)({\log }_{10}z-1)$
${\log }_{10}\left(zx\right)=2+({\log }_{10}z-1)({\log }_{10}x-1)$
such that $(x_1>x_2)$,
then match the elements of List - I with the correct answer in List -II.

$\begin{array}{cc}\text{List -I}& \text{List -II}\\ \left(I\right)\frac{y_1}{x_1}& \left(P\right)2\\ \left(II\right)\frac{z_1}{x_2}& \left(Q\right)100\\ \left(III\right)\frac{z_1{x}_2}{z_2}& \left(R\right)1000\\ \left(IV\right)\frac{y_2+z_1}{x_2}& \left(S\right)150\end{array}$

Which of the following is the only 'INCORRECT' combination?

1. $\left(I\right)\to \left(P\right)$
2. $\left(II\right)\to \left(Q\right)$
3. $\left(III\right)\to \left(S\right)$
4. $\left(IV\right)\to \left(S\right)$

Q. The value of ${\log }_5{\log }_3\sqrt{\sqrt[5]{59}}$ is

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

Q. If ${\log }_{x+1}(4x^3+9x^2+6x+1)+{\log }_{4x+1}(x^2+2x+1)=5,$ then the number of solution is

Q. The value of $4^{{\log }_{2}7}$ is

Q. Positive numbers $x,y$ and $z$ satisfy
$xyz$ = ${10}^{81}$ and $\left({\log }_{10}x\right)\left({\log }_{10}yz\right)+\left({\log }_{10}y\right)\left({\log }_{10}z\right)=468$, then the value of $({\log }_{10}x{)}^2+({\log }_{10}y{)}^2+({\log }_{10}z{)}^2$ is -

1. $5625$
2. $5652$
3. $5265$
4. $5526$

Q. If $lo{g}_{10}2=0.30103$, then $lo{g}_{10}50=$

1. 2.30103
2. 2.69897
3. 1.69897
4. 0.69897

Q. $log\{lo{g}_{ab}a+\frac{1}{lo{g}_{b}ab}\}=$

1. \N
2. 1
3. log ab
4. None of these

Q. The least integer greater than ${\log }_{2}15\cdot {\log }_{1/6}2\cdot {\log }_3\frac{1}{6}$ is

Q. How many real values of $x$ exist such that ${\log }_{(x+4)}(x^2-1)={\log }_{x+4}(5-x)$ holds good ?

1. $2$
2. $1$
3. Infinitely Many
4. $0$

Q. The value of $3^{\sqrt{{\log }_{3}4}}$ is equal to

1. $3^{\sqrt{{\log }_{4}3}}$
2. $4^{\sqrt{{\log }_{4}3}}$
3. $0$
4. $4^{\sqrt{{\log }_{3}4}}$