10-08-2021 Answers

1. The number of digits in 8^25 is

Answer: 23

Solution: We know log 2 = 0.301

8^25 = 2^75.

log 2^75 = 75*log2 = 75*0.301 = 22.575

Hence number of digits = Integer part of 22.575 + 1 = 22+1 = 23


2. The value of  log 0.000001 is

Answer: -6

Solution: log 0.000001 = log 10^-6 = -6


3. If log x = log 5 + 2 log 3 - 1/2 log 25, find the value of x.

Answer: 9

Solution: log x = log 5 + log 9 - log (25 ^ 1/2)

log x = log (5*9/5),

x = 9


4. If log(x+2) = log(x) + log(2), then find the value of x.

Answer: 2

Solution: log(x+2) = log(2x),

x+2 = 2x,

x=2


5. Find the value of log(0.125) 64

Answer: -2

Solution: 0.125 = 1/8

(1/8)^-2 = 64.

Hence log0.125 64 = -2


6. Find the value of log(25) 125 - log(8) 4

Answer: 5/6

Solution: log (5^2) 5^3 - log 2^3 2^2

= 3/2 - 2/3  = 5/6


7. If log(3) a = 4, find the value of a.

Answer: 81

Solution: a = 3^4 = 81


8. If log(10) 2 = 0.301, then the value of log(10) 50 is

Answer: 1.699

Solution: log 50 = log (100/2) = log 100 - log 2 = 2-0.301 = 1.699


9. If log 2 = 0.3010, log(2) 10 is equal to

Answer: 1000/301

Solution: log(2) 10 = log 10 / log 2 = 1/0.301 = 1000/301


10. If log 2 = 0.301 and log 3 = 0.4771, the number of digits in 12^12 is

Answer: 13

Solution: 12^12 = 2^24 * 3^12

Taking log,

log(2^24 * 3^12) = 24 log 2 + 12 log 4

= 24*0.301 + 12*0.4771 = 12.9492

Hence number of digits = 12 + 1 = 13


11. The value of log(2) 3 * log(3) 2 * log(3) 4 * log(4) 3 is

Answer: 1

Solution: The expression becomes (log3/log2) * (log2/log3) * (log4/log3) * (log3/log4)

= 1


12. If log 90 = 1.9542 then log 3 is equal to

Answer: 0.4771

Solution: log (9*10) = 1.9542,

log 9 + log 10 = 1.9542,

log (3^2) + 1 = 1.9542,

2 log3 = 0.9542,

log3 = 0.4771


13. If log(x-2) = log(x) - log(2), then find the value of x.

Answer: 4

Solution: log(x-2) = log(x/2),

x-2 = x/2,

x=4


14. The value of  log(9) 81 - log(4) 32 is

Answer: -1/2

Solution: log(9) 81 - log(4) 32 = log(9) 9^2 - log(2^2) 2^5

= 2 * 1 - 2.5

= 2 - 2.5

= -1/2


15. If log(10) x = 7, the value of x is

Answer: 10^7

Solution:  log10 x = 7,

 So x = 10^7


16. If log(x) 4 is 0.4, then the value of x is

Answer: 32

Solution: log(x) 4 = 0.4 = 2/5

log4/logx = 2/5,

2log2/logx = 2/5,

logx = 5 log2. Hence x = 2^5 = 32


17. Find the value of log 8 + log 1/8

Answer: 0

Solution: log 8 + log 1/8 = log (8*1/8) = log 1 = 0


18. If log 2 = 0.3010, log 5 is equal to

Answer: 0.699

Solution: log 5 = log (10/2) = log 10 - log 2 = 1 - 0.301 = 0.699


19. Find the number of digits in 2^47 (given log 2 = 0.3010)

Answer: 15

Solution: log 2^47 = 47 log 2 = 47*0.301 = 14.147

Number of digits = Integral part in 14.147 + 1 = 14+1 = 15


20. Find the number of digits in 8^10, given that log 2 = 0.301

Answer: 10

Solution: 8^10 = 2^30

30*0.301 = 9.03

Hence number of digits = Integral part of (9.03) + 1 = 9+1 = 10

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