Correct Answer: 1

Explanation:

Exp. =log₃₂2⁸ + log₂₄₃3⁷ - log₃₆1296
= log₂⁵ 2⁸ + log₃⁵ 3⁷ - log₃₆ 36²
= (8/5)log₂2 + (7/5)log₃3 - 2log₃₆36
= 8/5 + 7/5 - 2 = 1

Correct Answer: (1+2m)/2

Explanation:

log 49 28 = 1 2 log 7 7 × 4

 

= 1 2 + 1 2 2 log 7 2
= 1 2 + log 7 2
=  1 2 + m
=  1 + 2 m 2.

Correct Answer: 4

Explanation:

log 2 3 144 = 4 

Correct Answer: 13

Explanation:

 Let    L e t x = 3 3 3=  3 3 3

 

 Then,  log x = 3 3 log 3  

 

= 27 x 0.47712 = 12.88224 

 

Since the characteristic in the resultant value of log x is 12

 

?The number of digits in x is (12 + 1) = 13 

 

Hence the required number of digits in  3 3 3is 13.

Correct Answer: 4

Explanation:

 

Hence the least value of is 4

Correct Answer: -1/2

Explanation:

Rewrite equation as  1 2 2 x + 1 = 1 2 0

 

Leads to 2x + 1 = 0 

 

Solve for x : x = -1/2

Correct Answer: p > q

Explanation:

k l > k + 1 l + 1 for (k,l) > 0 and  k > l 

 

 

 

Let     k = x+1    and   l = x

 

 

 

Therefore,  x + 1 x > ( x + 1 ) + 1 ( x ) + 1

 

 

 

 (x + 1) > x

 

 

 

Therefore,  log ( x + 1 ) log ( x ) > log ( x + 2 ) log ( x + 1 )

 

 

 

? log x x + 1 > log x + 1 x + 2

Correct Answer: 3/2

Explanation:

  $$\begin{gathered} = log 3 3 + log 2 3 + log 10 3 log 10 × 3 × 2 2 \\ \end{gathered}$$ 

 

 

 

= log 3 3 1 2 + log 2 3 + log 10 3 1 2 log ( 10 × 3 × 2 2 )  

 

 

 

 

 

 

 

= 1 2 log 3 3 + 3 log 2 + 1 2 log 10 3 log 10 + log 3 + log 2 2  

 

 

 

 

 

 

 

= 3 2 log 3 + 2 log 2 + log 10 log 3 + 2 log 2 + log 10 =  3 2  

Correct Answer: 0

Explanation:

= log tan 1 0 + log tan 89 0 + log tan 2 0 + log tan 88 0 + ? ? + log tan 45 0  

 

= log [ tan 1 0 × tan 89 0 ] + log [ tan 2 0 × tan 88 0 ] + ? ? + log 1  

 

? tan ( 90 - ? ) = c o t ? a n d tan 45 0 = 1  

 

= log 1 + log 1 +.....+log 1 

 

= 0.

Correct Answer: 1