Overview: Logarithm | General Aptitude for GATE - Mechanical Engineering PDF Download

In Mathematics, logarithms are the other way of writing the exponents. A logarithm of a number with a base is equal to another number. A logarithm is just the opposite function of exponentiation. For example, if 102 = 100 then log10 100 = 2.
Hence, we can conclude that,
Logb x = n  or   bn = x
Where b is the base of the logarithmic function.
This can be read as “Logarithm of x to the base b is equal to n”.

History

John Napier introduced the concept of Logarithms in the 17th century. Later it was used by many scientists, navigators, engineers, etc for performing various calculations which made it simple. In simple words, Logarithms are the inverse process of exponentiation.

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What are Logarithms?

A logarithm is defined as the power to which a number must be raised to get some other values. It is the most convenient way to express large numbers. A logarithm has various important properties that prove multiplication and division of logarithms can also be written in the form of logarithm of addition and subtraction.

“The logarithm of a positive real number a with respect to base b, a positive real number not equal to 1[nb 1], is the exponent by which b must be raised to yield a”.
i.e. b^y = a  ⇔ log_ba = y
Where,

• “a” and “b” are two positive real numbers
• y is a real number
• “a” is called argument, which is inside the log
• “b” is called the base, which is at the bottom of the log.

In other words, the logarithm gives the answer to the question “How many times a number is multiplied to get the other number?”.
For example, how many 3’s are multiplied to get the answer 27?
If we multiply 3 for 3 times, we get the answer 27.
Therefore, the logarithm is 3.
The logarithm form is written as follows:
Log₃ (27) = 3 ….(1)
Therefore, the base 3 logarithm of 27 is 3.
The above logarithm form can also be written as:
3 x 3 x 3 = 27
3³ = 27 …..(2)
Thus, the equations (1) and (2) both represent the same meaning.
Below are some of the examples of conversion from exponential forms to logarithms.

Logarithm Types

In most cases, we always deal with two different types of logarithms, namely

• Common Logarithm
• Natural Logarithm

Common Logarithm

The common logarithm is also called the base 10 logarithms. It is represented as log10 or simply log. For example, the common logarithm of 1000 is written as a log (1000). The common logarithm defines how many times we have to multiply the number 10, to get the required output.

For example, log (100) = 2

If we multiply the number 10 twice, we get the result 100.

Natural Logarithm

The natural logarithm is called the base e logarithm. The natural logarithm is represented as ln or loge. Here, “e” represents the Euler’s constant which is approximately equal to 2.71828. For example, the natural logarithm of 78 is written as ln 78. The natural logarithm defines how many we have to multiply “e” to get the required output.
For example, ln (78) = 4.357.
Thus, the base e logarithm of 78 is equal to 4.357.

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Logarithm Rules and Properties

There are certain rules based on which logarithmic operations can be performed. The names of these rules are:

• Product rule
• Division rule
• Power rule/Exponential Rule
• Change of base rule
• Base switch rule
• Derivative of log
• Integral of log

Let us have a look at each of these properties one by one

Product Rule

In this rule, the multiplication of two logarithmic values is equal to the addition of their individual logarithms.
Logb (mn)= logb m + logb n
For example: log3 ( 2y ) = log3 (2) + log3 (y)

Division Rule

The division of two logarithmic values is equal to the difference of each logarithm.
Log_b (m/n)= log_b m – log_b n
For example, log₃ (2/ y) = log₃ (2) -log₃ (y)

Exponential Rule

In the exponential rule, the logarithm of m with a rational exponent is equal to the exponent times its logarithm.
Log_b (mⁿ) = n log_b m

Example 1: log_b(2³) = 3 log_b 2

Change of Base Rule

Log_b m = log_a m/ log_a b    
Example 2: logb 2 = loga 2/loga b

Base Switch Rule

logb (a) = 1 / loga (b)
Example3: log_b 8 = 1/log₈ b

Derivative of log

If f (x) = log_b (x), then the derivative of f(x) is given by;
f'(x) = 1/(x ln(b))

Example 4: Given, f (x) = log₁₀ (x)
Then, f'(x) = 1/(x ln(10))

Integral of Log

∫log_b(x)dx = x(log_b(x) – 1/ln(b)) + C

Example 5: ∫ log₁₀(x) dx = x ∙ (log₁₀(x) – 1 / ln(10)) + C

Other Properties

Some other properties of logarithmic functions are:

• Log_b b = 1
• Log_b 1 = 0
• Log_b 0 = undefined

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Logarithmic Formulas

log_b(mn) = log_b(m) + log_b(n)
log_b(m/n) = log_b(m) – log_b(n)
Log_b (xy) = y log_b(x)
Log_bm√n = log_b n/m
m log_b(x) + n log_b(y) = log_b(x^myⁿ)
log_b(m+n) = log_b m + log_b(1+nm)
logb(m – n) = log_b m + log_b (1-n/m)

Example 1: Solve log₂ (64) =?
Sol: since 2⁶ = 2 × 2 × 2 × 2 × 2 × 2 = 64, 6 is the exponent value and log₂ (64)= 6.

Example 2:  What is the value of log₁₀(100)?
Sol: In this case, 10² yields you 100. So, 2 is the exponent value, and the value of log₁₀(100) = 2

Example 3:  Use of the property of logarithms, solve for the value of x for log₃ x = log₃ 4 + log₃ 7
Sol: By the addition rule, log₃ 4+ log₃ 7= log₃ (4 * 7 )
Log₃ (28). Thus, x = 28.

Example 4: Solve for x in log₂ x = 5
Sol: This logarithmic function can be written In the exponential form as 2⁵ = x
Therefore, 2⁵ = 2 × 2 × 2 × 2 × 2 = 32, x = 32.

Example 5: Find the value of log5 (1/25).
Sol: Given:  log₅ (1/25)
By using the property,
Log_b (m/n)= log_b m – log_b n
log₅ (1/25) = log₅ 1 – log₅ 25
log₅ (1/25) = 0 – log₅ 5²
log₅ (1/25) = -2log₅5
log₅ (1/25) = -2 (1)  [By using the property log_a a = 1)
log₅ (1/25) = -2.
Hence, the value of  log₅ (1/25) = -2