Solved Examples: Logarithms

# 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

logb (x) if and only if by = 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: logb (xy) = logb (x) + logb (y)
• Quotient Rule: logb (x/y) = logb (x)−logb (y)
• Power Rule: logb  (x)n = nlogb (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."

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

### Exponential form

• 2= 8
• 4= 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 : px = qy
(a) z/x
(b) y/x
(c) X/z
(d)A/x
Ans:
(b)
log px = log qy
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 91/2)/  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 logx (5/18) = 1/2 , then find the value of x:
(a) 456/12
(b) 25/324
(d) 566/18
Ans:
(b)
logx5/18 = 1/2
x1/2 = 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 : logx√3 = 1/2
(a) 3
(b) 2
(c) 4
(d) 6
Ans: (a)
On evaluating the given equation,
x > 0 , x 1
x1/2 = √3
x1/2 = √3
squaring both side
(x1/2)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

The document Solved Examples: Logarithms | Quantitative Aptitude for SSC CGL is a part of the SSC CGL Course Quantitative Aptitude for SSC CGL.
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## FAQs on Solved Examples: Logarithms - Quantitative Aptitude for SSC CGL

 1. What is a logarithm and how is it related to exponents?
Ans. A logarithm is the inverse operation of exponentiation. It helps us solve equations where the unknown is in the exponent. For example, if we have the equation 2^x = 8, we can rewrite it as log base 2 of 8 = x, which tells us that 2 raised to the power of x equals 8.
 2. How do logarithms simplify calculations?
Ans. Logarithms simplify calculations by converting multiplicative operations into additive operations. For instance, if we have the equation log base 10 of (a * b) = log base 10 of a + log base 10 of b, we can break down the multiplication of a and b into the addition of their logarithms. This simplification makes complex calculations involving large numbers or exponential growth more manageable.
 3. What are the properties of logarithms?
Ans. Some important properties of logarithms include: - Product Rule: log base b of (a * c) = log base b of a + log base b of c - Quotient Rule: log base b of (a / c) = log base b of a - log base b of c - Power Rule: log base b of a^c = c * log base b of a - Change of Base Formula: log base b of a = log base c of a / log base c of b These properties allow us to manipulate logarithmic equations and solve complex problems.
 4. How can logarithms be used in real-life applications?
Ans. Logarithms have various applications in real-life scenarios, such as: - Sound Intensity: Logarithms are used to measure sound intensity through the decibel scale. - pH Scale: Logarithms are used to measure the acidity or alkalinity of a substance on the pH scale. - Earthquakes: Logarithms are used to measure the intensity of earthquakes using the Richter scale. - Compound Interest: Logarithms are utilized in financial calculations involving compound interest. These are just a few examples of how logarithms are applied in everyday life.
 5. Can logarithms be negative?
Ans. Logarithms can be negative, but only in certain cases. Logarithms are undefined for negative inputs and zero. However, when dealing with complex numbers or logarithms of negative inputs, we can use the properties of complex logarithms to obtain a negative result. In most cases, logarithms are positive or zero.

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