Important Formulas: Logarithms

# Important Formulas: Logarithms | Quantitative Aptitude for SSC CGL PDF Download

## Logarithm

A logarithm is a mathematical function designed to determine the exponent to which a particular base must be elevated to achieve a specified number. It stands as the reciprocal operation of exponentiation.

### General Formula of Logarithm

• where b is the base of the logarithm.
• y is is the number for which we want to find the exponent.
• x is the exponent to which b must be raised to get y.

### Definition & Logarithmic Formulas

Logarithms represent the exponent to which a number must be raised to attain another specified number.

## Types of Logarithms

There are two types:

• Common logarithm
• Natural logarithm

### Common Logarithm

Logarithm with a base of 10 is referred to as the common logarithm and is expressed as log10 X. If the base is not specified, it is assumed to be 10.
(a) Natural Logarithm:
(b) Logarithm with a base of 'e' is known as the natural logarithm and is expressed as loge X.

Crucial Note: In the absence of a specified base, always assume the base to be 10

### Formulas for Logarithm

• logx X= 1
• loga 1= 0
• a loglogax =X
• loga(x p) = p(log ax)
• loga (xy)= logaX+ logaY

Value of log(2 to 10): Remember

• Log 2 = 0.301
• Log 3 = 0.477= 0.48
• Log 4 = 0.60
• Log 5 = 0.698 = 0.7
• Log 6 = 0.778 = 0.78
• Log 7 = 0.845 = 0.85
• Log 8 = 0.90
• Log 9 = 0.954= 0.96
• Log 10 = 1

### Logarithm Formulas (Antilog):

• An antilog is the inverse function of a logarithm.
log(b) x = y means that antilog (b) y = x.
• The best way to understand any problem is by having a look at the Solved Example.
• We are going to do the same here, and we are going to understand the Antilog problem by Solved Example.

### Examples

Example 1: If log 27= 1.431, then the value of log 9 is?
(a) 0.945
(b) 0.934
(c) 0.958
(d) 0.954
Ans: (d)
log 27= 1.431
⇒ log(3)3= 1.431
⇒ 3log 3= 1.431
⇒ log3= 0.477
therefore, log9= log 32= 2 log3= (2×0.477)= 0.954

Example 2: Solve the equation log x= 1- log(x-3)
(a) 2
(b) 1/2
(c) 5
(d) 4
Ans: (c)
By combining both the equation we get
logx + log (x-3)=1
log(x(x-3))= log 101
Now convert it into exponential form,
x (x-3)= 101
x2 – 3x-10= 0
(x-5) (x+2)=0
x= -2, x=5
By solving this equation we get two values for x.
x= -2, x=5
Put the different value of x in different equation and solve them,
x= -2
log(-2) = 1- log (-2-3)
x= 5
log5 = 1-log(5-3)
log5 = 1-log2
Negative value is not considered in logarithm. So, we have a single value of x i.e, x=5.

Example 3: If log 10 5+ log(5x+1) = log 10 (x+5) +1, Find the value of X?
(a) 3
(b) 1
(c) 10
(d) 5
Ans: (a)
log 10 5+ log(5x+1) = log 10 (x+5) +1.
log 10 5+ log(5x+1) = log 10 (x+5) +log 10 10
log10 [5 (5x+1) ] = log10( 10 (x+5)]
5 (5x+1) = 10 (x+5)
5x+1 = 2x+ 10
3x= 9
x=3.

Example 4: If log   log(a+b), then
(a) a-b=1
(b) a=b
(c) a+b=1
(d) a2-b2 = 1
Ans: (c)

so, a+b=1

Example 5: log9 (3log2 (1+log3 (1+2log2 x)))= 1/2. Find x.
(a) 2
(b) 1/2
(c) 1
(d) 4
Ans: (a)

log2(1+log3(1+2log2 x) = 1
1+log3 (1+2log2 x)= 21
log3 (1+2log2 x)=2− 1
log3 (1+2log2 x) = 1
(1+2log2 x) = 31
1+ 2log2 x= 3
2log2 x = 2
log2x = 1
x= 2

The document Important Formulas: 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 Important Formulas: Logarithms - Quantitative Aptitude for SSC CGL

 1. What is a logarithm?
Ans. A logarithm is a mathematical function that represents the inverse operation of exponentiation. It helps to solve equations involving exponential functions by converting them into simpler forms.
 2. What are the types of logarithms?
Ans. There are two commonly used types of logarithms: natural logarithms (base e) and common logarithms (base 10). Natural logarithms are denoted as ln(x), while common logarithms are denoted as log(x) or log10(x).
 3. What are some important formulas related to logarithms?
Ans. Some important formulas related to logarithms are: - Logarithm of a product: log(ab) = log(a) + log(b) - Logarithm of a quotient: log(a/b) = log(a) - log(b) - Logarithm of a power: log(a^b) = b * log(a) - Change of base formula: log(base a) x = log(base b) x / log(base b) a
 4. How can logarithms be used in real-life applications?
Ans. Logarithms have various applications in real-life scenarios, such as: - Measuring the intensity of earthquakes using the Richter scale - Calculating pH levels in chemistry - Analyzing population growth and decay in biology - Assessing sound intensity levels in acoustics - Solving exponential growth and decay problems in finance and economics
Ans. Some frequently asked questions about logarithms in exams may include: - Solve the equation log(x) + log(x + 8) = 2 for x. - Simplify the expression log(4x) - log(2x). - Evaluate log(base 5) 25. - Solve the exponential equation 2^(x+1) = 8^(x-2). - Find the value of x in the equation 3^(2x-1) = 9^(x+2).

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