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1

WB JEE 2010

MCQ (Single Correct Answer)

The value of $${{{{\log }_3}5 \times {{\log }_{25}}27 \times {{\log }_{49}}7} \over {{{\log }_{81}}3}}$$ is

A
1
B
6
C
$${2 \over 3}$$
D
3

Explanation

$${{{{\log }_3}5 \times {{\log }_{{5^2}}}{3^3} \times {{\log }_{{7^2}}}7} \over {{{\log }_{{3^4}}}3}}$$

$$ = {{{3 \over 2}.\,{1 \over 2}} \over {{1 \over 4}}}\,.\,{{{{\log }_3}5\,.\,{{\log }_5}3\,.\,{{\log }_7}7} \over {{{\log }_3}3}} = 3$$

2

WB JEE 2009

MCQ (Single Correct Answer)

If x = logabc, y = logbca, z = logcab, then the value of $${1 \over {1 + x}} + {1 \over {1 + y}} + {1 \over {1 + z}}$$ will be

A
x + y + z
B
1
C
ab + bc + ca
D
abc

Explanation

$$\because$$ $$x = {\log _a}bc$$

$$ \Rightarrow 1 + x = {\log _a}a + {\log _a}bc = {\log _a}abc$$

Similarly, $$1 + y = {\log _b}abc$$ and $$1 + z = {\log _c}abc$$

$$\therefore$$ $${1 \over {1 + x}} + {1 \over {1 + y}} + {1 \over {1 + z}} = {1 \over {{{\log }_a}abc}} + {1 \over {{{\log }_b}abc}} + {1 \over {{{\log }_c}abc}}$$

$$ = {\log _{abc}}a + {\log _{abc}}b + {\log _{abc}}c = {\log _{abc}}abc = 1$$

3

WB JEE 2009

MCQ (Single Correct Answer)

The value of $$\left( {{1 \over {{{\log }_3}12}} + {1 \over {{{\log }_4}12}}} \right)$$ is

A
0
B
1/2
C
1
D
2

Explanation

$${1 \over {{{\log }_3}12}} + {1 \over {{{\log }_4}12}} = {\log _{12}}3 + {\log _{12}}4$$ ($$\because$$ $${\log _b}a = {1 \over {{{\log }_a}b}}$$)

$$ \Rightarrow {\log _{12}}12 = 1$$

4

WB JEE 2008

MCQ (Single Correct Answer)

If $${\log _5}\,\,{\log _5}\,\,{\log _2}x = 0$$ then value of x is

A
32
B
125
C
625
D
25

Explanation

$${\log _5}({\log _5}({\log _2}x)) = 0$$

$$ \Rightarrow {\log _5}({\log _5}({\log _2}x)) = {\log _5}1$$

$$ \Rightarrow {\log _5}({\log _2}x) = 1 = {\log _5}5$$

$$ \Rightarrow {\log _2}x = 5$$ (If $${\log _2}x = {\log _x}y \Rightarrow x = y$$)

$$ \Rightarrow x = {2^5} = 32$$.

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