Difference between Ln and Log: Definitions and Properties

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Difference between ln and log is that ln is defined for base e while log is defined for base 10 i.e., loge = ln. Ln is also called natural log. In mathematics, logarithm is defined as the inverse function of exponentiation. In simpler words, logarithm is the power (exponent) to which a base has to be raised to attain a given number i.e., the number of times a base should be multiplied by itself to attain a given number. Logarithms help to cover a large range of data/values in a crisp manner. Due to this reason, it has numerous applications in the field of mathematics, physics and chemistry.

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Read More: Trigonometry Table


Definition of Ln

Ln is known as the natural logarithm i.e., logarithm with the base e (an irrational and a transcendental number) which is approximately equal to 2.7183. It is referred to by ln x or loge x. Here, e is the Euler’s constant.

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loge x = ln x = ex = y

A table representing the ln values of 1 to 10 is:

ln (1) 0
ln (2) 0.6931
ln (3) 1.0986
ln (4) 1.3863
ln (5) 1.6094
ln (6) 1.7918
ln (7) 1.9459
ln (8) 2.0794
ln (9) 2.1972
ln (10) 2.3026

Rules of Ln

  • Power Rule:

Ln (xy) = y * ln (x)

Example: ln (53) = 3 * ln (5)

  • Quotient Rule:

Ln (x/y) = ln (x) – ln (y)

Example: ln (7/3) = ln (7) – ln (3)

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  • Reciprocal Rule:

Ln (1/x) = - ln (x)

Example: ln (1/5) = - ln (5)

Properties of natural Log
Properties of natural Log

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Definition of Log

Log is known as the logarithm which is defined for the base 10. It is also called common logarithm. 

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Tan2x Formula

Log10 y = x

10x = y

A table representing the log values of 1 to 10 is:

log (1) 0
log (2) 0.3010
log (3) 0.4771
log (4) 0.6020
log (5) 0.6989
log (6) 0.7781
log (7) 0.8450
log (8) 0.9030
log (9) 0.9542
log (10) 1

Properties of Log:

  • Log (mn) = log m + log n

Example: log (3*5) = log (3) + log (5)

  • Log (m/n) = log m – log n

Example: log (3/2) = log (3) – log (2)

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  • Log (mn) = n log m

Example: log (52) = 2 * log (5)

  • Logb m = loga m/loga b

Example: log3 8 = log2 8 * loga 2

Properties of Log
Properties of Log

Difference between Ln and Log

S. No. Ln Log
1. Ln is defined as the logarithm with base e Log is defined as the logarithm with base 10
2. It is also known as natural logarithm It is also known as common logarithm
3. It is represented as loge x It is represented as log10 x
4. Its exponential form is ex = y Its exponential form is 10x = y
5. General statement for exponential logarithm is “By what number must e be raised to attain y” General statement for exponential logarithm is “By what number must 10 be raised to attain y”
6. It is mathematically represented as log base e It is mathematically represented as log base 10
7. It finds less application in physics It finds more application in physics

Application of Ln and Log

The real life applications are:

  • Logarithms are used in the Richter Scale for earthquakes.
  • It is also used in the detection of the password strength
  • For data scientists, logarithms help in seeing the patterns in bulk data
  • Log odds play a pivotal role in logistic regression

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Points to Remember

  • The general form of a logarithm is:

loga (y) = x

The above expression can also be denoted as:

ax = y

  • loge x = ln x = ey = y
  • loga 1 = 0
  • loga a = 1
  • Ln (x) = 2.303 * log (x)
  • Log (x) = ln (x) / 2.303

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Sample Questions

Ques 1. Define logarithms. [1 mark]

Ans. Logarithm of a given number with base x is the power to which x should be raised to attain the given number.

ax = y

loga y = x

Ques 2. Define natural and common logarithms. [2 mark]

Ans. Natural logarithm is a logarithm with the base e(Euler’s constant which is approximately equal to 2.7183) i.e., loge x = ln x = ex

Common logarithm is a logarithm with the base 10 i.e., log10 x

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Ordinate?

Collinear points?

Ques 3. What is the value of log 9 when log 27 is equal to 1.431? [3 mark]

Ans. Given, 

Log 27 = 1.431

Log (33) = 1.431

3 * log 3 = 1.431

Log 3 = 1.431/3

Log 3 = 0.447

Log 9 = log (32)

Log 9 = 2 * log 3

Log 9 = 2 * 0.447

Log 9 = 0.954 

Ques 4. Given that the value of log10 2 = 0.3010, what is the value of log10 80? [3 mark]

Ans. Log10 80 = log10 (8 * 10)

Log10 8 + log10 10

Log10 (23) + 1

3 * log 2 + 1

3 * 0.3010 + 1

0.9030 + 1

Hence, the value of log10 80 is 1.9030

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Ques 5. If the value of log10 5 + log10 (5x + 1) = log10 (x + 5) + 1, what is the value of x? [4 mark]

Ans. Given,

log10 5 + log10 (5x + 1) = log10 (x + 5) + 1

 log10 5 + log10 (5x + 1) = log10 (x + 5) + log10 10

 log10 [5 (5x + 1)] = log10 [10(x + 5)]

 5(5x + 1) = 10(x + 5)

 5x + 1 = 2x + 10

 3x = 9

Hence, the value of x is 3.

Ques 6. If the value of log10 7 = a, then log10 (1/70) is equal to? [3 mark]

Ans. log10 (1/70) = log10 1 – log10 70

= - log10 (7 * 10)

= - (log10 7 + log10 10)

= - (a + 1)

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Ques 7. How can you convert natural log to common log? [2 mark]

Ans. To convert a natural logarithmic value to common logarithmic value, the following formula can be used:

Ln (x) = log (x) ÷ log (2.71828)

Ques 8. Find the value of (1/log3 60 + 1/log4 60 + 1/log5 60) [3 mark]

Ans. Given,

(1/log3 60 + 1/log4 60 + 1/log5 60)

= log60 (3 * 4 * 5)

= log60 (60)

Hence, the answer to this question is 1.

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CBSE CLASS XII Related Questions

  • 1.
    If \( f(x) = \begin{cases} \frac{\sin^2 ax}{x^2}, & \text{if } x \neq 0 \\ 1, & \text{if } x = 0 \end{cases} \) is continuous at \( x = 0 \), then the value of 'a' is :

      • 1
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    • 2.
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        • 3.
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            • 4.
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                  • 6.

                    Based upon the results of regular medical check-ups in a hospital, it was found that out of 1000 people, 700 were very healthy, 200 maintained average health and 100 had a poor health record.
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                    and \( A_3 \): People with poor health.
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