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Logarithm MCQ

1. Which one is correct?

(a)    log (m + n) = log m + log n

(b)    log (m – n) = log m – log n

(c)    log_b a x log_a b =1

(d)    None of these.

log_b a x log_a b = log_a a x log_b b = 1 x 1 = 1. So, option (c) is correct

2. Find the value of log _b a . log_c b . log_a c

(a)    abc

(b)    0

(c)    1

(d)    4.

log_b a. log_c b. log_a c = (log_c a. log_b b). log_a c = (log_c a x 1) log_a c = log_c a  x  log_a c = log_a a. log_c c = 1 x 1 = 1. So, option (c) is correct

3. If log 2 = 0.30103, then the value of log 200 is

(a)    18.0103

(b)    6.30103

(c)    4.30103

(d)    2.30103

log 200 = log (2 x 100) = log 2  +  log 100 = log 2  +  2 log 10 = 0.30103 + 2 X 1 = 2.30103

(we know log 2 = 0.30103, log 10 = 1). So, option (d) is correct

4. log 144 is equal to

(a)    2 log 4 + 2 log 2

(b)    4 log 2 + 2 log 3

(c)    3 log 2 + 4 log 3

(d)    None of these.

• 2 log 4 + 2 log 2 = log 4² + log 2² = log 16 + log 4 = log (16 x 4) = log 64
• 4 log 2 + 2 log 3 = log 2⁴ + log 3² = log 16 + log 9 = log (16 x 9) = log 144
• 3 log 2 + 4 log 3 = log 2³ + log 3⁴ = log 8 + log 81 = log (8 x 81) = log 648

So, option (b) is correct

5. The relation log_e x + log_e (1 + x) = 0 is equal to

(a)    x² + x + e = 2

(b)    x² + x – e = 3

(c)    x² + x – 1 = 0

(d)    x² + x + 4 = 0.

log_e x + log_e (1 + x) = 0, Or,  log_e x(1 + x) = 0, Or, x(1 + x) = e⁰, Or, x + x²  = 1, Or, x² + x – 1 = 0. So, option (c) is correct

_{6. Find value of  (½)log10 (25) – 2 log10 (3) +log10 (18)}

(a)      0

(b)     1

(c)      3

(d)     2.

 log₁₀[an1]  25 _{– 2 log10  3 + log10 18= log 10 25} ^½ _{–  log 10 3}² _{+log 10 18 = log 10 5 –  log 10 9 +log 10 18}

_{(log 10 5 –  log 10 9) + log 10 18 = log 10 (5/9) + log 10 18 = log 10 [5 X 18)/9]} = _{log 10} 10 = 1.

So, option (b) is correct

7. If log₁₀ 2 = 0.3010, the value of log₁₀ 80 is

(a)    1.9030

(b)    1.6020

(c)    2.9030

(d)    4.9030.

Log₁₀ 80=log₁₀ (8 X 10) = log₁₀  8  + log₁₀ 10 = log₁₀  2³ + 1 = 3₁₀ log  2 + 1

= 3 x 0.3010 + 1 = 0.9030 + 1 = 1.9030. So, option (a) is correct

8. Find the value of log₃ (27)³

(a)    27

(b)    9

(c)    729

(d)    81.

log₃ (27)³ = 3 log₃ (27)= 3 log₃ 3³    = (3 X 3) log₃ 3  = 9 log₃³ = 9  X1 = 9 (we know log₃ ³ = 1).

So, option (b) is correct

9. If log₃ (log₂ x) = 1, Find the value of x

(a)    10

(b)    9

(c)    3

(d)    8

log₃ (log₂ x) = 1, or, log₃ (log₂ x) = 1, Now, log₂ x = 3¹ = 3, Or, x = 2³ = 8, So, option (d) is correct

10. If log₄ (x² + x) – log₄ (x + 1) = 2, then:

(a)    x = 8

(b)    x = 4                                          

(c)    x = 6                                          

(d)    x = 16.                              

log₄ (x² + x) – log₄ (x + 1) = 2,  or, log₄ [(x² + x) / (x + 1)] =2. Or, [(x² + x) / (x + 1)] = 4² = 16

Or, x²+ x = 16(x + 1), Or, x²+ x = 16x + 16, Or, x²+ x – 16x – 16 = 0 , Or, x(x +1) – 16(x +1) = 0

Or, (x +1) (x -16), Either   x = -1  or,  x = 16, So, option (d) is correct

11. Value of  (log₁₀ 125 / log₁₀ 25) = ?

a. 5

b.

c.

d.

(log₁₀ 125 / log₁₀ 25) = (log₁₀ 5³ / log₁₀ 5²) = 3 log₁₀ 5 / 2 log₁₀ 5 = 3/2. So, option (b) is correct

12. If log₈ m + log₈ 2=2/3, then find the value of m

(a)    1

(b)    3

(c)    2

(d)    4

log₈ m + log₈ 2=2/3, or, log₈ (m X 2) + =2/3, or, log₈ 2m =2/3,

So, 8 ^(2/3) = 2m. or, {2³}^(2/3) = 2m. Or, 2² = 2m, Or, m = 2. So, option (c) is correct

13. Value of 7 log₁₀ (16/15)+ 5 log₁₀ (25/24)+ 3 log₁₀ (81/80)

(a)    2

(b)    log₁₀ 2

(c)    log₁₀ 3

(d)    log₁₀ 5

7 log₁₀ (16/15)+ 5 log₁₀ (25/24)+ 3 log₁₀ (81/80)

= 7 {log₁₀ 16 –  log₁₀15)} + 5 {log₁₀ 25 – log₁₀  24} + 3 [{ (log₁₀  3⁴) – log₁₀ (80}]

= 7 {log₁₀ 2⁴ –  log₁₀ (5 X 3)} + 5 {log₁₀ 5² – log₁₀ (8X3)} } + 3 { log₁₀  (3⁴)- log₁₀ (2⁴ X 5)}

= 7 { 4log₁₀ 2 – ( log₁₀ 5 + log₁₀ 3)} + 5 {2 log₁₀ 5- (log₁₀ 2³ + log₁₀3) } + 3 { 4log₁₀  3  – (4log₁₀ 2 + log₁₀ 5)}

= 28 log₁₀  2 – 7 log₁₀  5- 7 log₁₀ 3 +10 log₁₀  5 – 15 log₁₀ 2 – 5 log₁₀ 3 + 12 log₁₀  3 – 12 log ₁₀ 2- 3 log ₁₀ 5

= log ₁₀ 2. So, option (b) is correct

14. If log₁₀ p + log₁₀q = log₁₀ (p + q), which one of the following relations is true?

(a)    p=(q/q-1)

(b)    3p=q=0

(c)    3p=q²/(1-q)

(d)    p=q=2

log₁₀   p + log₁₀  q = log₁₀  (p + q), Or,  log₁₀   (pq) = log₁₀  (p + q), Or,  pq = p + q, Or,  pq – p = q,

Or,  p(q – 1) = q, or, p=(q/q-1). So, option (a) is correct

15. If log [(x+y)/5] = ½ (logx + log y), find value of  (x/y) + y/x)

(a)       20

(b)       23

(c)       25

(d)       18.

log [(x+y)/5] = ½ (logx + log y), or, log [(x+y)/5] = ½ log (xy), or,  log [(x+y)/5] = ½ log (xy),

log [(x+y)/5] = log [(xy) ^½ ].  Or, (x+y)/5 = (xy) ^½, or {(x+y)/5}² = (xy), or, (x² + y² + 2xy) / 25=xy

or, (x² + y² + 2xy) =25xy,  x² + y²=23xy, or, (x² + y²) / xy =23,

Now, (x/y) + y/x) = (x² + y²) / xy = 23. So, option (b) is correct

16. If log (2a – 3b) = log a – log b, then a =

(a)       3b² / (2b – 1)

(b)       3b³  /  (2b – 1)

(c)       b²/[4 (2b + 1)]

(d)       None of the above

log (2a – 3b) = log a – log b, or, log (2a – 3b) = log (a/b). or, 2a-3b=a/b, Or, 2ab – 3b² = a

Or, 2ab – a = 3b², Or, a(2b – 1) = 3b², or a= 3b²/ (2b – 1). So, option (a) is correct

17. Find value of log (a²/bc) + log (b²/ac) + log (c²/ab)

(a)    1

(b)    0

(c)    3

(d)    log a

log (a²/bc) + log (b²/ac) + log (c²/ab)

= (log a² – log bc) + (log b² – log ac) + (log c² – log ab)

= log a² + log b² + log c² – (log bc + log ac + log ab)

= log a² b² c² – log (bc. .ab)

= log a² b² c²  – log a² b² c²  = 0. So, option (b) is correct

18. If log (m/n) + log (n/m) = log (m+n) then find true statement

(a)      m+n=1

(b)     m/n=2

(c)      m – n = 2

(d)     m² – n² = 4

log (m/n) + log (n/m) = log (m+n), or log {(m/n) X (n/m)} = log (m+n), or log1=log (m+n). or m+n=1. So, option (a) is correct

19. If log_a3= 1/3, then find value of a

(a)    27

(b)    81

(c)    9

(d)    24.

log_a3= 1/3, So, a^1/3=3, or a=(3)³ = 27. So, option (a) is correct

20. Find the value of log₂ (1/64)

(a)    6

(b)    -6

(c)    64

(d)    4.

log₂ (1/64) = log₂ (1/2⁶) =  log₂ 1 – 6log₂ 2 = 0 – (6X 1) = -6 (as  log₂ 2= 1). So, option (b) is correct

21. log_x(16/25) = -1/2, Find value of x

(a)    625/256

(b)    256/625

(c)    526/265

(d)    32/50

log_x(16/25)=-1/2. Or x^(-1/2) =16/25, or  1/{x^(1/2)} =16/25, or x^(1/2)} = 25/16, or x=25²/16²= 625/256. So, option (a) is correct

22. If log_x10000 = -1/4, find x

(a)	1/10	√
(b)	1/100
(c)	1/20
(d)	1/1000

log_x10000 = -1/4, so, (10000)^-1/4= x, or 1/ (10000)^1/4 = x, or x= 1/{(10⁴)}^1/4 = 1/10.

So, option (a) is correct

23. Find value of log₁₀ (0.00001)

(a)      – 5

(b)     1/5

(c)      – 4

(d)     5

log₁₀ (0.00001) = log₁₀ (1/100000) = log₁₀ (1/10⁵) = log₁₀ 10^-5) = 5 log₁₀ = -5 X1= – 5 .

So, option (a) is correct

24. If log₁₀ x + log₁₀y = z, then find value of x

(a)    z/y

(b)    10/3z

(c)    10^z/y

(d)    yz

log₁₀   x + log₁₀  y = z, or Or, log₁₀   (xy) = z,  10^z = xy, x=10^z/y. So, option (c) is correct

25. Find value of log₁₀ 5 assuming log₁₀ 2 = 0.3010

(a)    .4010

(b)    0.6911

(c)    0.6990

(d)    .6021.

log₁₀  5 = log₁₀ (10/2) = log₁₀ 10 – log₁₀ 2 = 1- 0.3010 =0.6990. So, option (c) is correct

26. Find value of (log₅ 3) x (log₃ 625)

(a)    1

(b)    2

(c)    3

(d)    4.

log₅  3 x log₃  625 = log ₃ 3 x log₅  625 (by change of base) = 1 x log₅  5⁴ = log₅  5⁴ = 4 log₅ 5 = 4X1=4. So, option (d) is correct

27. If log(x² – 6x + 10) = 0, then the value of x is

(a)    4

(b)    7

(c)    3

(d)    5.

log  (x² – 6x + 10) = 0, Or, log( x² – 6x + 10) = log 1, x² – 6x + 10=1, or  x² – 6x + 10 – 1 = 0  

Or, x² – 6x + 9 = 0, Or, x² – 3x – 3x + 9 = 0, Or, x (x – 3) – 3(x – 3) = 0, Or, (x – 3) (x – 3) = 0, or x = 3. So, option (c) is correct

28. Find x, if log₄3 + log₄(x + 2) = 2

(a)    10/3

(b)    4/3

(c)    2/3

(d)    12/5

log₄   3 + log₄ (x + 2) = 2,  log₄   {3 X (x+2)} =2, or, log₄   {3x+6)} =2,

Or, 4² = 3x + 6  Or, 3x+ 6 = 16 Or, 3x = 16- 6 = 10 Or, x = 10/3. So, option (a) is correct

_{29.  If a}² _{+ b}² _{= 23ab, evaluate log {(a+b)/5}}

(a)    5

(b)    log(ab) /2

(c)    log(ab) /5

(d)    log(a+b) /4

a² + b² = 23ab, Or, a² + b² + 2ab = 23ab + 2ab, Or, (a + b)² = 25ab, {(a+b)/5)}²=ab, or (a+b/5)=(ab) ^½  

or log {(a+b/5)}=log {(ab) ^½ } = ½ log ab = {log(ab)} /2. So, option (b) is correct

30.Evaluate {1/log_ab(abc)} + {1/log_bc(abc)} + {1/log_ca(abc)}

(a)     0 (b)     1 (c)     2 (d)     3

{1/log_ab(abc)} + {1/log_bc(abc)} + {1/log_ca(abc)}

= log abc (ab) +  log abc (bc) +  log abc (ca) = log abc (ab . bc . ca), = log abc (abc)2

= 2  log _abc abc = 2.1 = 2. So, option (c) is correct

31. If log (2a – 3b) = log a – log b, find value of a

(a)    3b²/(2b – 1)

(b)    3b/2b – 1

(c)    2b²/(4b + 2)

(d)    2b²/2b – 1

log (2a – 3b) = log a – log b, Or, log (2a – 36) = log (a/b), Or, 2a – 3b = a/b

Or, 2ab – 3b² = a, Or, 2ab – a = 3b², Or, a(2b – 1) = 3b², or a=3b² / (2b – 1). So, option (a) is correct

32. If log (a+b/4) = 1/2 (log a + log b), then find value of (a/b + b/a)

(a)    18 (b)    14 (c)    15 (d)    10

log (a+b/4) = 1/2 (log a + log b), or log (a+b/4) = log (ab)^1/2, or, (a+b/4) =(ab)^1/2,

or {(a+b)2}/16 = ab, Or, a2 + 2ab + b2 = 16ab, or, a2 + b2 = 14ab, (a2 + b2) / (ab) = 14, (a/b + b/a) =14. So, option (b) is correct