1.If a^x = b^y, then

a.log a/b= x/y     b.log a/ log b = x/y    c.log a/ log b = y/x    d.log b/a = x/y

2.2 log₁₀ 5 + log₁₀ 8 – ½ log₁₀ 4 = ?

a.2            b.4                   c.2 + 2 log₁₀ 2                        d.4 - 4 log₁₀ 2

3.log_a (ab) = x, then log _b (ab) is :

a.1/x         b.x/ (x+1)                    c.x/(1-x)          d.x/(x-1)

4.If log₈ x + log ₈ 1/6 = 1/3, then the value of x is:

a.12         b.16                c.18                 d.24

5.The value of (log₉ 27 + log₈ 32) is:

a.7/2        b.19/6             c.5/3                d.7

6.If log₁₂ 27 = a, then log₆ 16 is:

a.(3-a)/4(3+a)   b.(3+a)/4(3-a)            c.4(3+a)/(3-a)            d.4(3-a)/(3+a)

7.The value of (1/log₃ 60  + 1/log₄ 60  + 1/log₅ 60) is:

a.0           b.1                   c.5                   d.60

8.If log x + log y = log (x+y),then,

a.x=y               b.xy=1          c.y= (x-1)/x                 d.y=x/(x-1)

9.If  log 27= 1.431, then the value of log 9 is:

a.0.934              b.0.945           c.0.954           d.0.958

10.If  log 2= 0.030103, the number of digits in 2⁶⁴ is :

a.18              b.19                c.20                 d.21

Answer & Explanations

1.(c). a^x = b^y => log a^x = log b^y => x log a = y log b

=> log a/ log b = y/x

2.(a). 2 log₁₀ 5 + log₁₀ 8 – ½ log₁₀ 4

= log₁₀ (5²) + log₁₀ 8 -  log₁₀ (4^1/2)

= log₁₀ 25 + log₁₀ 8  - log₁₀ 2 = log₁₀ (25*8)/2

= log₁₀ 100 = 2

3.(d). log_a (ab) = x => log b/ log a = x => (log a + log b)/ log a = x

1+ (log b/ log a) = x => log b/ log a = x-1

log a/ log b = 1/ (x-1) => 1+ (log a/ log b) = 1 + 1/ (x-1)

(log b/ log b) + (log a/ log b) = x/ (x-1) => (log b + log a)/ log b = x/ (x-1)

=>log (ab)/ log b = x/(x-1) => log_b (ab) = x/(x-1)

4.(a). log₈ x + log₈ (1/6) = 1/3

=> (log x/ log 8) + (log ¹/₆ / log 8) = log (8^1/3) = log 2

=> log x = log 2 – log 1/6 = log (2*6/1)= log 12

5.(c). Let log₉ 27 = x. Then, 9^x = 27

=> (3²)^x = 3³ => 2x = 3 => x= 3/2

Let log₈ 32 = y. Then

8^y = 32 => (2³)^y = 2⁵ => 3y = 5 => y=5/3

6.(d). log₁₂ 27 = a => log 27/ log 12 = a

=> log 3³ / log (3 * 2²) =a

=> 3 log 3 / log 3 + 2 log 2 = a => (log 3 + 2 log 2)/ 3 log 3 = 1/a

=> (log 3/ 3 log 3) + (2 log 2/ 3 log 3) = 1/3

=> (2 log 2)/ (3 log 3) = 1/a – 1/3 = (3-a)/ 3a

=> log 2/ log 3= (3-a)/3a => log 3 = (2a/3-a)log2

log₁₆ 16 = log 16/ log 6 = log 2⁴/ log (2*3) = 4 log 2/ (log 2 + log 3)

= 4(3-a)/ (3+a)

7.(b). log₆₀ 3 + log₆₀ 4 + log₆₀ 5 + log₆₀ (3*4*5)

= log₆₀ 60 = 1

8.(d). log x + log y = log (x+y)

=> log (x+y) = log (xy)

=> x+y = xy => y(x-1) = x

=> y= x/(x-1)

9.(c). log 27 = 1.431 => log 3³ = 1.431

=> 3 log 3= 1.431 => log 3 = 0.477

Therefore, log 9 = log 3² = 2 log 3 = (2*0.477) = 0.954

10.(c). log 2⁶⁴ = 64 log 2 = (64*0.30103) = 1926592

Its characteristics is 19.

      Hence, the number of digits in 2⁶⁴ is 20.