Test Level 1: Exponents and Logarithm - CAT MCQ

log 3 = 0.47712 …. (1)
log (364) = 64 x  log 3
= 64 x 0.47712 (from equation1)
= 30.53568
Its characteristic is 30.
Hence, the number of digits in 364 is (30 + 1) = 31
Option (3) is correct.

log10 x - 1/2 log10 x = 2 logx 10 ...(I)
or, 1/2 log10 x = 2 logx 10 ...(II)
Using base change rule (logb a = 1/loga b), equation (II) becomes:
1/2 log10 x = 2/log10 x
(log10 x)² = 4
or, log10 x = ±2
x = 100 or x = 1/100

p^a = q^b = r^c = k (Assume)

Given: 
⇒ q² = pr

3 = 12^z/x ... (1)
4 = 12^z/y ... (2)
Multiply (1) and (2);
12 = 12^{(z/x) + (z/y)}

log₄ ³ × log₅ ⁴ x … x log₉ ⁸ × log₃ ⁹

(Comparing this with a - b √3 : a = 11, b = 6)

log10 75 = log10 (3 x 5²) = log10 3 + 2log10⁵
log₁₀ ²⁵² = log10 (2² x 3² x 7) = 2 log10 ² + 2log10 ³ + log10 ⁷
log₁₀ ⁹⁸ = log10 (7² x 2) = 2 log10 ⁷ + log10 ²
All of the above can be easily solved, but log10 770 (log10 ⁷ x log₁₀ 11) cannot be solved because log₁₀ 11 is unknown.
Hence, option (d) is correct.

log2 x² + logx 2 = 3
⇒ + log_x 2 = 3
⇒ 2 + (logx 2)² - 3 log_x 2 = 0
Put y = logx 2
⇒ y² - 3y + 2 = 0
⇒ y = 1, 2
For y = 1, logx 2 = 1 or x = 2
For y = 2, logx 2 = 2 or x = √2
Hence, there are two valid values of x, i.e. x = 2, √2.

Given sequence can be written as follows.
log7 2, log7 2², log7 2⁴, log7 2⁸, ……….
It can be further written as
log₇ 2, 2 log₇ 2, 4 log7 2, 8 log7 2, ………. [log mⁿ = n log m]
By dividing 2nd term by 1st term, we get common ratio, r = = 2