Solved Examples: Logarithms | Quantitative Aptitude for SSC CGL PDF Download

Definition

A logarithm represents the exponent to which a number must be raised to yield another number. The majority of aptitude questions related to logarithms rely on formulas.

Logarithms Formula

log_b (x) if and only if b^y = x

Common Logarithm and Natural Logarithm 

Common logarithm has base 10 (b = 10) and is denoted as log(x), while natural logarithm has base e (Euler’s number) and is denoted as ln(x).

Logarithm Rules

• 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)ⁿ = nlog_b (x)

A logarithm is expressed as the inverse of exponentiation. In simpler terms, when we take the logarithm of a specific value, we essentially reverse the process of exponentiation. Here are discussions and answers related to logarithm questions:
For example: If we choose a base such as b = 3 and raise it to the power of k = 2, the result is 32, denoted by C, expressed as 32 = C. The rules of exponentiation can be applied to deduce that the result is C = 32 = 8.
As an illustration, consider someone asking, "To which power must 2 be raised to equal 16?" The answer is 4. This can be expressed through logarithmic calculation, i.e., log2(16) = 4, which is articulated as "log base two of sixteen is four."

Logarithm form

• Log₂(8) = 3
• Log₄(64) = 3
• Log₅(25) = 2

Exponential form

• 2³ = 8
• 4³ = 64
• 5²= 25

Generalizing the examples above leads us to the formal definition of a logarithm.
Logb (a) =c ↔ bc =a
Both the equations define the similar link where:
‘b’ is considered as the base,
c is considered as the exponent
a is considered as the argument

Examples

Example 1: If the value of log 3 = 0.477, then find the number of digits in 3³⁶
(a) 18
(b) 3
(c) 20
(d) 24
Ans: (b)
Number of digits in 336 ≈ ⌊log10(336)⌋ + 1
Number of digits in 336 ≈ ⌊log(112 * 3)⌋ + 1
Number of digits in 336 ≈ ⌊(log(112) + log(3))⌋ + 1
Number of digits in 336 ≈ ⌊(2.049 + 0.477)⌋ + 1
Number of digits in 336 ≈ ⌊2.526⌋ + 1
Number of digits in 336 ≈ 2 + 1
Number of digits in 336 ≈ 3
So, the number 336 has 3 digits.

Example 2: The value of log₃ 27 is :
(a) 3
(b) 2
(c) 7
(d) 8
Ans: (a)
log₃ 3³
= 3

Example 3: Solve : p^x = q^y 
(a) z/x
(b) y/x
(c) X/z
(d)A/x
Ans: (b)
log p^x = log q^y
x log p = y log q
log p / log q = y/x

Example 4: Solve: log √9 /log 9
(a) zero
(b) 1/2
(c) 1
(d) 1/3
Ans: (b)
(log 9^1/2)/  log 9
= 1/2(log 9/ log 9)
= 1/2

Example 5: Solve: log₆x³ = 18
(a) 46656
(b) 4/9
(c) 223456
(d) 4/6
Ans: (a)
x³ = 6¹⁸
x = 6⁶
x = 46656

Example 6: If log_x (5/18) = 1/2 , then find the value of x:
(a) 456/12
(b) 25/324
(c) 324/78
(d) 566/18
Ans: (b)
log_x5/18 = 1/2
x^1/2 = 5/18
√x = 5/18
Squaring both sides
x = (5/18)²
x = 25/324

Example 7: If log 5 = 0.698, find the number of digits in 5²⁵
(a) 3
(b) 18
(c) 4
(d) 6
Ans: (b)
log (5²⁵) = 25* log 5
= 25* 0.698
= 17.45
= 18 (approx.)

Example 8: Solve the given logarithmic equation: log₇x = 3
(a) 324
(b) 343
(c) 289
(d) 366
Ans: (b)
Taking base 7 antilog on both sides,
log₇x = 3
⇒ X = 7³
⇒ X = 343

Example 9: Solve : log_x√3 = 1/2
(a) 3
(b) 2
(c) 4
(d) 6
Ans: (a)
On evaluating the given equation,
x > 0 , x 1
x^1/2 = √3
x^1/2 = √3
squaring both side
(x^1/2)² = √3²
x = 3

Example 10: Prove : log₆36  = 3x
(a) 2/3
(b) 4/5
(c) 6/7
(d) 6/8
Ans: (a)
log₆36 = log₆(6)² 
2log₆6 = 2 ( logaa = 1)
2 = 3x
x = 2/3