Fundamental Laws of Logarithms

Q. $\text{If}y=\sqrt{x+\sqrt{x+\sqrt{x+\cdots \infty }}}$, then $\frac{dy}{dx}=$

1. $\frac{1}{y^2-1}$
2. $\frac{1}{2y+1}$
3. $\frac{2y}{y^2-1}$
4. $\frac{1}{2y-1}$

Q. Let $A=\{n\in N|n^2\le n+10,000\},$ $B=\{3k+1|k\in N\}$ and $C=\left\{2k|k\in N\right\}$. Then the sum of all the elements of the set $A\cap (B-C)$ is equal to

Q.

What is the value of $\log \left(\frac{1}{2}\right)$?

Q. $7log\left(\frac{16}{15}\right)+5log\left(\frac{25}{24}\right)+3log\left(\frac{81}{80}\right)$ is equal to

1. \N
2. 1
3. log 2
4. log 3

Q. If $a={\log }_{245}175$ and $b={\log }_{1715}875,$ then the value of $\frac{1-ab}{a-b}$ is

Q. If $lo{g}_{12}27=a,thenlo{g}_{6}16=$

1. $2.\frac{3-a}{3+a}$
2. $3.\frac{3-a}{3+a}$
3. $4.\frac{3-a}{3+a}$
4. $Noneofthese$

Q. The value of ${\log }_{10}5\cdot {\log }_{10}20+({\log }_{10}2{)}^2$ is

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

Q. The value of ${\log }_5{\log }_2{\log }_3{\log }_{2}512$ is

Q. The value of $lo{g}_{3}4lo{g}_{4}5lo{g}_{5}6lo{g}_{6}7lo{g}_{7}8lo{g}_{8}9$ is

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

Q. If ${\log }_3(3x-8)=2-x,$ then the value of $x$ is

1. $-1$
2. $2$
3. $9$
4. $3$

Q. The value of $\frac{1}{{\log }_{3}60}+\frac{1}{{\log }_{4}60}+\frac{1}{{\log }_{5}60},$ is

Q. If $lo{g}_4$ 5 = a and $lo{g}_5$ 6 = b, then $lo{g}_3$ 2 is equal to

1. $\frac{1}{2a+1}$
2. $\frac{1}{2b+1}$
3. $\frac{1}{2ab-1}$
4. $2ab+1$

Q. Consider the four sets $A,B,C$ and $D$ defined as
$A=\{a:a=8({\log }_4\left(x\right){)}^3+4{\log }_{\sqrt{2}}\left(x^4\right),x>0\}$
$B=\{b:b=20+\frac{13}{({\log }_{x}2{)}^2},x>0\}$
$C=\{c\in N:c\in A\cap B\text{for same}x>0\}$
$D=\{x\in Z:(x-2{)}^2(15x^2-56x+17)<0\}$

Then the number of elements in $C\times D$ is

Q. The value of $\sqrt{{\left({\log }_{0.5}4\right)}^2}$ is

Q. If $3^a=4,4^b=5,5^c=6,6^d=7,7^e=8$ and $8^f=9,$ then the value of the product $\left(abcdef\right)$ is

Q.

What is the value of $\ln \left(10\right)$?

Q.

The sum of the integers from $1$ to $100$ which are divisible by $3$ and $5$, is

1. $2317$

2. $2632$

3. $315$

4. $2489$

Q. ${\log }_{n}1+{\log }_n(1+\frac{1}{2})+{\log }_n(1+\frac{1}{3})+\dots \dots +{\log }_n(1+\frac{1}{n-1})=$

1. $1-{\log }_{n}2$
2. $1+{\log }_{n}2$
3. ${\log }_2(n-1)$
4. ${\log }_n(n+1)$

Q. Given that ${\log }_{10}2=0.3010$, the number of digits in the number ${2000}^{2000}$ is

1. $6600$
2. $6603$
3. $6543$
4. $2345$

Q. If $a^{1-2x}.b^{1+2x}=a^{4+x}.b^{4-x};a,b>0\&a\ne b$, then the value of $x$ is

1. ${\log }_{\left(\frac{b}{a}\right)}\left(ab\right)$
2. ${\log }_{\left(\frac{a}{b}\right)}\left(ab\right)$
3. $\frac{1}{2}{\log }_{\left(\frac{a}{b}\right)}\left(ab\right)$
4. $\frac{1}{2}{\log }_{\left(\frac{b}{a}\right)}\left(ab\right)$

Q.

Expand $\log 15.$

Q. If $\log (x+z)+\log (x-2y+z)=2\log (x-z)$, then

1. $y(x+z)=2xz$
2. $z(z+y)=2xy$
3. $x^2-z^2=y$
4. $x+z=3y$

Q. The value of $\frac{2\log 2+\log 3}{\log 48-\log 4}$ is

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

Q. If $x=lo{g}_a\left(bc\right),y=lo{g}_b\left(ca\right),z=lo{g}_c\left(ab\right),$ then which of the following is equal to 1

1. x + y + z
2. $(1+x{)}^{-1}+(1+y{)}^{-1}+(1+z{)}^{-1}$
3. xyz
4. None of these

Q.

If $A=\left[\begin{array}{cc}2& -3\\ -4& 1\end{array}\right]$, then $adj(3A^2+12A)$is equal to:

1. $\left[\begin{array}{cc}51& 63\\ 84& 72\end{array}\right]$

2. $\left[\begin{array}{cc}51& 84\\ 63& 72\end{array}\right]$

3. $\left[\begin{array}{cc}72& -63\\ -84& 51\end{array}\right]$

4. $\left[\begin{array}{cc}72& -84\\ -63& 51\end{array}\right]$

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 number of real solution(s) of the equation ${\left(x^{{\log }_{10}3}\right)}^2-3^{{\log }_{10}x}=2$ is

Q. The equation $x^{({\log }_{3}x{)}^2-\frac{9}{2}{\log }_{3}x+5}=3\sqrt{3}$ has

1. at least one real root
2. exactly one real root
3. exactly one irrational root
4. complex roots

Q. If $lo{g}_{10}2=0.30103,lo{g}_{10}3=0.47712$ the number of digits in $3^{12}\times 2^8$ is

1. 9
2. 10
3. 7
4. 8