Q: If log (x + 4) = log(4) + log(x) and log (x + 6) = log (6) then which of the following correct?

Given log (x + 4) = log(4) + log(x) ⇒ x + 4 = 4x ∴ x = 4/3 Similarly y = 5/4 ∴ x > y

Question 1

Find the number of digits in 8 10? (Given that log 10 2 = 0.3010)

Accepted Answer

810 = (23)10 ∴ Required answer = [30 log10 2 + 1] = [30 x 0.3010] + 1 = 9.03 + 1 = 9 + 1 = 10

Question 2

Find the no. of digits in 8 57 (given that log 102 = 0.3010)

Accepted Answer

857 = (23)57 = 2171 ∴ Required answer = (171 log10 2 + 1 ) = [171 x 0.3010] + 1 = [51.4710] +1 = 51+1 =52

Question 3

If log(x - 5) = log(x) - log(5) and log(y - 6) = log(y) - log(6) then which of the following is correct?

Accepted Answer

Given log( x - 5) = log(x) - log(5) ⇒ x -5 = x/5 ∴ x = 25/ 4 ....(i) Again from question log(y - 6) = log(y) - log(6) ⇒ y - 6 = y/6 ∴ y = 36/5 ...(ii) From equations (i) & (ii) x < y

Question 4

If log (x + 4) = log(4) + log(x) and log (x + 6) = log (6) then which of the following correct?

Accepted Answer

Given log (x + 4) = log(4) + log(x) ⇒ x + 4 = 4x ∴ x = 4/3 Similarly y = 5/4 ∴ x > y

Question 5

If a x = b, b y = c, c z = a, then the value of xyz is?

Accepted Answer

∵ ax = b ⇒ loga b = x ∵ by = c ⇒ logb c = y ∵ cz = a ⇒ logc a = z ∴ xyz = logab x logbc x logca = 1

Question 6

If log xy = 100 and log 2x = 10 then the value of y is?

Accepted Answer

logxy =100, log2x = 10 ⇒ log y / log x = 100 and log x / log 2 = 10 ⇒ log y / log 2 = 100 x 10 = 1000 ⇒ log2y = 1000 ∴ y = 21000

Question 7

If log x4 = 0.4 then the value of x is?

Accepted Answer

logx4 = log 4 / log x = 2/5 ⇒ 2log2 / log x = 2/5 ⇒ log x =5log 2 = log 25 ⇒ log x = log 32 ∴ x = 32

Question 8

If log 1227 =a then log 616 is?

Accepted Answer

∵ log1227 = a ⇒ log 27 / log 12 = a ⇒ a log 12 = log 33 ⇒ a log ( 3 x 4 ) = 3 log 3 ⇒ a[log 3 + log 4] = 3 log 3 ⇒ a log 4 + a log 3 = 3 log 3 ⇒ a log 22 = ( 3 - a) log 3 ⇒ 2a log 2 = (3 - a) log 3 ∴ log 2 / log 3 = (3 - a) / 2a Now log616 = log16 / log 6 = log 24 / log (2 x 3) = 4 log 2 / ( log 2 + log 3) = [4 (log 2 / log 3)] / [(log 2 / log 3) + 1] = 4[(3 - a) / 2a] / [{(3 - a) / 2a } + 1] = 4(3 - a) / (3 + a)

Question 9

If log (0.57) = 1756, then the value of log 57 + log (0.57) 3 + log √ 0.57 is?

Accepted Answer

Given Exp. = log (57 x 100 / 100) + 3log(0.57) + 1/2log(0.57) = log (0.57) + log 102 + 3log(0.57) +1/2log (0.57) = (1 + 3 + 1/2) log(0.57) + 2 [∵ log102 = 2] = (4.5 x 1.756) + 2 = 4.5 x (-1 + 0.756) + 2 = 3.402 - 4.5 + 2 = 0.902

Question 10

(log tan1°. log tan2° ............. log tan50°) is?

Accepted Answer

∵ log tan 45 = log 1 = 0 Hence, Whole Expression = (something) x zero

Question 11

The mantissa of log 3274 is 0.5150, then the value of log 32.74 is?

Accepted Answer

Since, 0.3274 gives characteristic 1. Therefore value of log (0.3274) = 1.5150

Question 12

If log 5 (x 2 + x) - log 5 x = 2 then the value of x is?

Accepted Answer

∵ log5[(x2 + x ) / x] = 2 ⇒ log10(x + 1) = 2 ⇒ x + 1 = 25 ∴ x = 24

Question 13

If log a, log b, log c are in A.P. then?

Accepted Answer

∵ loga, logb, logc are in A.P. Then, ⇒ logb - loga = logc - logb ⇒ log b/a = log c/b ⇒ b/a = c/b ⇒ b2 = ac ∴ a, b, c are in G.P.

Question 14

The compound interest on Rs. 12,000 for 10 years at the rate of 12% per annum compounded annually is?

Accepted Answer

∵ A = 12,000(1 + 12/100)10 = 12000(28/25)10 ⇒ log A = log 12000 + 10[log 28 - log 25] ⇒ log A = 4.0792 + 10(1.4472 - 1.3979) = 4.0792 + 0.493 = 4.5722 ∴ A = antilog 4.5722 = 37342 C.I. = 37342 - 12000 = 25342 =25350

Question 15

The number of digits in the numeral for 2 64?

Accepted Answer

∵ x = 264 ⇒ log x = log 264 ⇒ log x = 64 log 2 = 64 x .3010 = 19.264 ∴ No.of digits = 19 + 1 = 20

Question 16

Given log 10 2 = 0.30103, log 10 3 = 0.47712. Find the number of digit, in 3 12 x 2 8?

Accepted Answer

Let N = 312 x 28 ⇒ log N = 12 x log3 + 8 x log 2 ⇒ log N = 12 x 0.47712 + 8 x 0.30103 ⇒ log N = 8.13368 ∴ No.of digits = 8 + 1 = 9

Question 17

The value of 1/(log 2 π) + 1/(log 6 π) is?

Accepted Answer

Given exp. = 1/(log2 π) + 1/(log6 π) = logπ 2 + logπ 6 = logπ (2 x 6) = logπ12 Since 12 > π so the value of given expression is more than 1.

Question 18

The number of digits in the numeral for (875) 16?

Accepted Answer

∵ N = (875)16 ⇒ log N = log (875)16 ⇒ log N = 16 log (875) = 16 x (2.942) = 47.072 ∴ No.of digits = 47 + 1 = 48

Question 19

The population of a town at the beginning of the year 1986 was 2,65,000. If the rate of increase be 52 per thousand of the population. Find the population at the beginning of the year 1991.?

Accepted Answer

We have r = Rate of increase = 52/1000 x 100 = 5.2, n = 5, P0 = 265000 ∵ P = 265000(1 + 5.2 / 100)5 ⇒ log P = log 265000 + 5(log 105.2 - log 100) = 5.4232 + 5(2.0220 - 2) = 5.4232 + 0.1100 = 5.5332 ∴ P = antilog(5.5332) = 341400

Question 20

What rate per cent per annum compound interest will Rs. 3,000 in 3 years if the interest is reckoned half yearly?

Accepted Answer

∵ 3000 = 2000(1 + r/200)6 ⇒ 3/2 = (1 + r/200)6 ⇒ 1 + r/200 = (3/2)1/6 ⇒ log(1 + r/200)= 1/6(log 3 - log 2 ) ⇒ log(1 + r/200)= 1/6(0.4771-.3010) = 0.02935 ⇒ (1 + r/200) = antilog(.02935) ⇒ 1 + r/200 = 1.070 = 1 + 7/100 ∴ r = 14%

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