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Last Updated: Aug 31, 2023

# Difference between Log and Ln(Natural Log)

## Introduction

Hello, ninjas! Logarithmic functions are mainly used to solve equations where an unknown variable is an exponent of any other quantity. Many branches of mathematics and subjects use logarithms to solve complex problems.

Logarithmic functions are basically in two forms, a Common Log and another Natural Log.

In this article, we will cover the difference between Log and Ln(Natural Log) briefly, and we will also see some examples of these functions and their properties.

## What is Logarithm (Log)

The inverse function of the exponentiation is known as the Logarithm. We can also define the Log as the power of a number that must raise a number to obtain the other number. It is the Logarithm of base 10 or common Logarithm.

We can write Logarithms in general form as

Loga(b) = c

This can be written as:

ac = b

### Some Properties of Logarithm

• Loga(xy) = Logax + Logay
• Loga(x/y) = Logax - Logay
• Loga(xy) = y Logax
• Logb(x) = Loga(x) / Loga(b)
• Loga(b) = 1 / Logb(a)
• Loga(a) = 1

### Examples of these properties

#### Example 1

Find x,  Log5(x) = 2.

Solution:

We know,

Logab = c can be written as ac = b

So,

Log5x = 2 is equivalent to x = 52

Since, x = 25 Ans

#### Example 2

Using the property Loga(x) = Logb(x) / Logb(a), Prove Loga(x) = 1 / Logx(a).

Solution:

In a given property, put b = x

So, Loga(x) = Logx(x) / Logx(a)

We know, Logxx = 1,

Loga(x) = 1 / Logx(a)

Hence Proved.

## What is Natural Logarithm (Ln)

Natural Logarithm or Ln is the Logarithm with base e. Here e is a constant, an irrational and transcendental number whose value is approximately equal to 2.718281828459…

We can represent the natural Logarithm as ln or Loge (read as Log with base e).

All the common logarithmic properties will also be valid in Natural Logarithms.

### Examples of Natural Logarithm

#### Example 1

Find x, Ln(x) = Ln(6) + 2Ln(5) - Ln(3).

Solution:

We know, aLog(b) = Log(ba)

so , 2Ln(5) = Ln(52) = Ln(25)

Now,

Ln(x) = Ln(6) + Ln(25) - Ln(3)

Also, Log(a) + Log(b) = Log(ab) and Log(a) - Log(b) = Log(a / b)

Ln(x) = Ln(6 * 25) - Ln(3)

Ln(x) = Ln(150 / 3) = Ln(50)

Since, Ln(x) = Ln(50)

So, x = 50 Ans

#### Example 2

Find x, Ln(e) / Ln(43) = Log43(x).

Solution:

We know, Logaa = 1 and Logab = 1 / Logba

So, Ln(e) = 1

Now,

1 / Ln(43) = Log43(x)

Log43e = Log43(x)

Hence, x = e = 2.7182… Ans

## Difference between Log (Common Logarithm)and Ln (Natural Logarithm)

The difference between Log and Ln must be known to solve the problems related to them. A fundamental understanding of these logarithmic functions will help you to understand the various other concepts.

Some of the difference between Log and Ln are given below.

### How is Common Logarithm different from Natural Logarithm?

Common Logarithm is the Log with base 10 while Natural Logarithm is Ln with is Log with base e. Common Logarithm is represented as Log10(x), and Natural Logarithm is defined as Loge(x). This is the main difference between Log and Ln.

### What is the use of Logarithmic functions?

Logarithmic functions are used to solve equations where an unknown variable is an exponent of any other quantity. This makes it easy to solve the exponential questions quickly and with lesser calculations.

### Can negative numbers be used as input for Log and Ln?

Negative numbers cannot be used as input for Log and Ln functions. These functions are only defined for positive numbers and zero. We will get the complex number if we use the negative number as input.

### Can you convert Log to Ln or vice versa?

Yes, you may use the change of base formula to convert between Log and Ln. For example, divide the log value by 2.303 to convert log base 10 to Ln, and multiply the Ln value by 2.303 to convert Ln to Log base 10.

## Conclusion

In Conclusion, Logarithmic functions are crucial instruments for resolving exponential equations whose exponents contain unknown variables. The base is the primary difference between Log and Ln. As the base of Log is 10, and Ln is e.

We mainly use Log in physics compared to Ln, and Log and Ln are the common log and natural Log, respectively.

We hope this article helps you learn more about the Log functions and the difference between Log and Ln. You can practice more questions on this topic from Problems on logarithms for aptitude.