Logarithm: Aptitude Question and Answers

Logarithm is an important topic that needs to be prepared well for the Quantitative Aptitude section of exams in India. It requires practicing a lot of questions within time. Logarithmic formulas make it easy to solve questions easily in competitive examinations. The following article covers the concepts, formulas, and rules that a learner needs to know before approaching the questions.

Logarithmic function is inverse to the exponential function. A logarithm to the base b is the power to which b must be raised to produce a given number. For example, is equal to the power to which 2 must be raised in order to produce 8. Clearly, 2^3 = 8 so = 3. In general, for b > 0 and b not equal to 1. 

Logarithm Formulas

There are several logarithm formulas, but some of the most common ones are:

• Logarithmic identity:
  log(b^x) = x * log(b)
• Product rule:
  log(b, xy) = log(b, x) + log(b, y)
• Quotient rule:
  log(b, x/y) = log(b, x) – log(b, y)
• Power rule:
  log(b, x^p) = p * log(b, x)
• Change of base formula:
  log(b, x) = log(a, x) / log(a, b)

Where:

• b is the base of the logarithm
• x and y are the arguments of the logarithm
• p is a constant
• a is a different base, usually chosen to be 10 or e.

Sample Questions on Logarithm with Solutions:

Question 1: Find the value of x in equation given 8^x+1 – 8^x-1 = 63

Solution:

Take 8^x-1 common from the eq.
It reduce to
8^x-1(8² – 1) = 63
8^x-1 = 1
Hence, x – 1 = 0
x = 1

Question 2: Find the value of x for the eq. given log_0.25x = 16

Solution:

log_0.25x = 16
It can be write as
x = (0.25)¹⁶
x = (1/4)¹⁶
x = 4^-16

Question 3: Solve the equation log₁₂1728 x log₉6561

Solution:

It can be written as:
log₁₂(12³) x log₉(9⁴)
= 3log₁₂12 x 4log₉9
= 3 x 4 = 12

Question 4: Solve for x : log_x3 + log_x9 + log_x27 + log_x81 = 10

Solution:

It can be write as:
log_x(3 x 9 x 27 x 81) = 10
log_x(3¹ x 3² x 3³ x 3⁴) = 10
log_x(3¹⁰) = 10
10 log_x3 = 10
then, x = 3

Question 5: If log(a + 3 ) + log(a – 3) = 1 ,then a=?

Solution:

log₁₀((a + 3)(a – 3))=1
log₁₀(a² – 9) = 1
(a² – 9) = 10
a² = 19
a = √19

Question 6: Solve 1/log_ab(abcd) + 1/log_bc(abcd) + 1/log_cd(abcd) + 1/log_da(abcd)

Solution:

=log_abcd(ab) + log_abcd(bc) + log_abcd(cd) + log_abcd(da)
=log_abcd(ab * bc * cd * da)
=log_abcd(abcd)²
=2 log_abcd(abcd)
=2

Question 7: If xyz = 10 , then solve log(xⁿ yⁿ / zⁿ) + log(yⁿ zⁿ / xⁿ) + log(zⁿ xⁿ / yⁿ)

Solution:

log(xⁿ yⁿ / zⁿ * yⁿ zⁿ / xⁿ * zⁿ xⁿ / yⁿ)
= log xⁿ yⁿ zⁿ
= log(xyz)ⁿ
= log₁₀ 10ⁿ
= n

Question 8: Find (121/10)^x = 3

Solution:

Apply logarithm on both sides
log_(121/10)(121/10)^x = log_(121/10)3
x = (log 3) / (log 121 – log 10)
x = (log 3) / (2 log 11 – 1)

Question 9: Solve log(2x² + 17)= log (x – 3)²

Solution:

log(2x² + 17)= log (x² – 6x + 9)
2x² + 17 = x² – 6x + 9
x² + 6x + 8 = 0
x² + 4x + 2x + 8 = 0
x(x + 4) + 2(x + 4) = 0
(x + 4)(x + 2)=0
x= -4,-2

Question 10: log₂(33 – 3^x)= 10^{log(5 – x)}. Solve for x.

Solution:

Put x = 0
log₂(33 – 1)= 10^log(5)
log₂32 = 5
5 log₂ 2 = 5
5 = 5
LHS = RHS   

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