Fundamental Laws of Logarithms

Q.

What is the value of $e^{\infty }$?

Q.

One fourth of a herd of camel was seen in forest..Twice the square root of the herd had gone to mountain and the remaining 15 camels were seen on the banks of a river.Find the total number of camels.

Q.

If log (m+n)=log m+ log n , show that n=m/(m-1).

Q. $lo{g}_4$ 18 is

1. A rational number
2. An irrational number
3. A prime number
4. None of these

Q. The value of
$lo{g}_{3}2lo{g}_{4}3lo{g}_{5}4\dots lo{g}_{15}14lo{g}_{16}15$ is

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

Q. The value of $2^{lo{g}_{3}5}-5^{lo{g}_{3}2}$ is

1. 0
2. 3
3. 5
4. 2

Q. The value of $(0.05{)}_{\sqrt{20}}^{log}(0.1+0.01+0.001+.....)$

1. $\frac{1}{20}$
2. $\frac{1}{81}$
3. $81$
4. $20$

Q. If $x^{\frac{3}{4}(lo{g}_{3}x{)}^2+lo{g}_{3}x-\frac{5}{4}}=\sqrt{3}$ then x has

1. One positive integral value
2. One irrational value
3. Two positive rational values
4. None of these

Q. ${\sum }_{n=1}^n\frac{1}{lo{g}_{2^n}\left(a\right)}=$

1. $Noneofthese$
2. $\frac{(n+1{)}^2{n}^2}{4}lo{g}_{2}a$
3. $\frac{n(n+1)}{2}lo{g}_{a}2$
4. $\frac{n(n+1)}{2}lo{g}_{2}a$

Q. If $x^{18}=y^{21}=z^{28}then3,3lo{g}_{y}x,3lo{g}_{z}y,7lo{g}_{x}z$ are in

1. A.P.
2. H.P.
3. G.P.
4. None

Q. log ab - log |b| =

1. log a
2. -log a
3. None of these
4. log |a|

Q. If $lo{g}_a\left(ab\right)=x$, then $lo{g}_b\left(ab\right)$ is equal to

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

Q. If $a=lo{g}_{24}12,b=lo{g}_{36}24,c=lo{g}_{48}36$, then 1 + abc is equal to

1. 2 ac
2. 2 bc
3. None of these
4. 2 ab

Q. The value of $lo{g}_{9}27-lo{g}_{27}9$ is

1. $\frac{3}{2}$
2. 0
3. $\frac{5}{6}$
4. None

Q.

Value of x for log^x 100 - 2 log_x 2 = 2 is

1. -6

2. -4

3. 5

4. 4

Q. If n=1983!, then the value of expression $\frac{1}{lo{g}_{2}n}+\frac{1}{lo{g}_{3}n}+\frac{1}{lo{g}_{4}n}+\dots \frac{1}{lo{g}_{1983}n}$ is equal to

1. -1
2. 0
3. 1
4. 2

Q.

If $lo{g}_{a}b=lo{g}_{b}c=lo{g}_{c}a$, then a, b, c are such that __________ .

1. a = b = c

2. a = b ≠ c

3. a ≠ b = c

4. a = c ≠ b

Q. Find the value of $\frac{(0.6{)}^0-(0.1{)}^{-1}}{\left(\frac{3}{2^2}{)}^{-1}\right(\frac{3}{2}{)}^3+(\frac{-1}{3}{)}^{-1}}$

Q.

If n > 1, the value of $\frac{1}{lo{g}_{2}n}+\frac{1}{lo{g}_{3}n}+....+\frac{1}{lo{g}_{53}n}$ is

1. 1

2. $\frac{1}{lo{g}_{n!}53}$

3. None of these.

4. $\frac{1}{lo{g}_{53!}n}$

Q. Given:$\log 2=0.3010$ and $\log 3=0.4771$, find the value of $\log 25$.

Q. If $lo{g}_{10}2=0.30103$, then what is the value of $lo{g}_{10}50=$

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

Q. If $a=lo{g}_{24}12,b=lo{g}_{36}24andc=lo{g}_{48}36$ then 1+abc is equal to

1. 2ab
2. 2ac
3. 2bc
4. 0

Q.

The solution set of $2lo{g}_{2}lo{g}_{2}x+lo{g}_{1/2}lo{g}_2\left(2\sqrt{2x}\right)=1$ is

1. {2, 4}
   

2. {4, 8}
   

3. {1/2, 4}
   

4. {8}
   

Q. $\frac{1}{lo{g}_{xy}xyz}+\frac{1}{lo{g}_{yz}xyz}+\frac{1}{lo{g}_{zx}xyz}=$

1. 0
2. 1
3. 2
4. $lo{g}_{zx}xyz$

Q. Expand the following.
$\log 1000$

Q.

Fill in the boxes.

$\square +\frac{3}{4}=2\frac{22}{24}$

Q.

Question 128(i)

Predicting the ones digit, copy and complete this table and answer the questions that follow.
Powers Table
$\begin{array}{ccccccccccc}X& 1^x& 2^x& 3^x& 4^x& 5^x& 6^x& 7^x& 8^x& 9^x& {10}^x\\ 1& 1& 2& & & & & & & & \\ 2& 1& 4& & & & & & & & \\ 3& 1& 8& & & & & & & & \\ 4& 1& 16& & & & & & & & \\ 5& 1& 32& & & & & & & & \\ 6& 1& 64& & & & & & & & \\ 7& 1& 128& & & & & & & & \\ 8& 1& 256& & & & & & & & \\ \text{Ones digits of the powers}& 1& 2,4,6,8& & & & & & & & \end{array}$
Describe patterns you see in the ones digits of the powers.

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

1. $\sqrt{(-4)}$
2. None of these
3. -2
4. 2

Q.

Puja has two ropes measuring 5 m and 6 m. She ties a goat at point A with 5 m rope and ties another goat at point B with 6 m rope. Find out the ratio of areas grazed at point A and point B. The goats can only move inside the field.

1. 25:36

2. 26:37

3. 24:36

4. 25:37