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Difference Between Log and Ln

The difference between log and ln is that log is defined for base 10 and ln is denoted for base e. For example, log of base 2 is represented as logand log of base e, i.e. loge = ln (natural log).

A natural logarithm can be referred to as the power to which the base ‘e’ that has to be raised to obtain a number called its log number. Here e is the exponential function. It was initially discovered in the 17th century by John Napier, who discovered and conceptualized the theory of logarithms. Before looking into the key difference between ln and log, let’s understand the definition of log and ln.

Log and Ln Definition

Log: In Maths, the logarithm is the inverse function of exponentiation. In simpler words, the logarithm is defined as a power to which a number must be raised in order to get some other number. It is also called the logarithm of base 10, or common logarithm. The general form of a logarithm is given as:

loga (y) = x

The above-given form is written as:

ax = y

Rules of Logarithm: There are four major rules or properties of the logarithm.

  • Logb (mn)= logb m + logb n
  • Logb (m/n)= logb m – logbn
  • Logb (mn) = n logb m
  •  Logm = logm/ loga b

Ln: Ln is called the natural logarithm. It is also called the logarithm of the base e. Here, e is a number which is an irrational and transcendental number and is approximately equal to 2.718281828459… The natural logarithm (ln) is represented as ln x or logx

What are the Key Differences Between Log and Ln?

One must know the difference between log and ln to solve logarithmic problems. Having a fundamental understanding of the logarithm function can also prove beneficial to understanding different concepts. Some of the main differences between the natural logarithm and logarithm are given below:

Difference Between Log and Ln

Log

Ln

Log refers to a logarithm to the base 10

Ln refers to a logarithm to the base e

This is also called as a common logarithm

This is also called as a natural logarithm

The common log is represented as log10 (x)

The natural log is represented as loge (x)

The exponent form of the common logarithm is 10x =y

The exponent form of the natural logarithm is ex =y

The interrogative statement for the common logarithm is “At which number should we raise 10 to get y?” The interrogative statement for the natural logarithm is “At which number should we raise Euler’s constant number to get y?”
It is more widely used in physics when compared to ln

As logarithms are usually taken to the base in physics, ln is used much lesser

Mathematically, it is represented as log base 10 Mathematically, this is represented as log base e

Solved Examples

Example 1:

Solve for x, if log (3375)/ log 15 = log x.

Solution:

Given is a logarithmic function with base 10.

Now,  log (3375)/ log 15 = log x

⇒  log 153/ log 15 = log x

Using the property, loga bn = n loga b, we have

3 log 15/log 15 = log x

⇒ log x = 3

We know log 1000 = log 103 = 3 log 10 = 3

∴ x = 1000.

Example 2:

If s = e280 and t = e300, prove that ln (es2t –1) = 261.

Solution:

Given, s = e280 and t = e300

Taking natural logarithm on both sides, we get

ln (s) = ln (e280) = 280 and ln (t) = ln (e300) = 300

Now, ln (es2t –1) = ln e + ln (s2) + ln (t –1)

= 1 + 2 × ln (s) – ln (t)

= 1 + 2 × 280 – 300

= 1 + 560 – 300 = 261.

Example 3: 

Solve for x: 5x= 2e5.

Solution:

Taking natural logarithm on sides, we get,

ln (5x) = ln (2e5)

⇒ x ln 5 = ln 2 + 5 × ln e

⇒ x = (ln 2 + 5)/ln 5 = (0.693147 + 5)/1.609438

⇒ x = 3.5374 (approx.).

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Frequently Asked Questions on the Difference between Log and Ln

What is the difference between ln and log?

The difference between log and ln is that log is defined for base 10 and ln is denoted for base e.

What is a logarithm?

A logarithm is defined as the power to which number must be raised to get some other values.

What are the properties of a logarithm?

The properties of logarithms are:
log mn = log m + log n
Log (m/n)= log m – log n
Log (mn) = n log m
Log_b m = log_a m/ log_a b

What is natural logarithm?

The logarithmic function to the base e is called the natural logarithmic function and it is denoted by loge.
f(x) = loge x
Quiz on ln vs log

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