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 log₂ and log of base e, i.e. log_e = 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:

log_a (y) = x

The above-given form is written as:

a^x = y

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

• Log_b (mn)= log_b m + log_b n
• Log_b (m/n)= log_b m – log_bn
• Log_b (mⁿ) = n log_b m
•  Log_b m = log_a m/ log_a 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 log_e x

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:

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, log_a bⁿ = n log_a 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: 5^x= 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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