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:

Common Logarithm (log)

Natural Log (natural logarithm)

Log (Common Logarithm) generally refers to a logarithm to the base 10. Ln (Natural Logarithm) refers to a logarithm to the base e.

What is Log (Logarithm)?

Logarithm is the inverse function of exponentiation. 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

Log_{a}(b) = c

This can be written as:

a^{c} = b

Some Properties of Logarithm

Log_{a}(xy) = Log_{a}x + Log_{a}y

Log_{a}(x/y) = Log_{a}x - Log_{a}y

Log_{a}(x^{y}) = y Log_{a}x

Log_{b}(x) = Log_{a}(x) / Log_{a}(b)

Log_{a}(b) = 1 / Log_{b}(a)

Log_{a}(a) = 1

Examples of these properties

Example 1

Find x, Log_{5}(x) = 2.

Solution:

We know,

Log_{a}b = c can be written as a^{c} = b

So,

Log_{5}x = 2 is equivalent to x = 5^{2}

Since, x = 25 Ans

Example 2

Using the property Log_{a}(x) = Log_{b}(x) / Log_{b}(a), Prove Log_{a}(x) = 1 / Log_{x}(a).

Solution:

In a given property, put b = x

So, Log_{a}(x) = Log_{x}(x) / Log_{x}(a)

We know, Log_{x}x = 1,

Log_{a}(x) = 1 / Log_{x}(a)

Hence Proved.

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What is LN (Natural Logarithm)?

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 Log_{e} (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(b^{a})

so , 2Ln(5) = Ln(5^{2}) = 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) = Log_{43}(x).

Solution:

We know, Log_{a}a = 1 and Log_{a}b = 1 / Log_{b}a

So, Ln(e) = 1

Now,

1 / Ln(43) = Log_{43}(x)

Log_{43}e = Log_{43}(x)

Hence, x = e = 2.7182… Ans

Difference Between Log and Ln

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 differences between Log and Ln are given below.

Parameters

Logarithm (log)

Natural Logarithm (ln)

Base

Can have any base, commonly base 10 (log10) or base e (ln)

Base e (Euler's number, approximately 2.71828)

Common Use Cases

Engineering, physics, and general mathematics.

Mathematics, especially calculus and exponential functions.

Symbol

Typically denoted as log(base, value)

Denoted as ln(value)

Base Conversion

Can be converted to different bases using change of base formula.

Doesn't require conversion as it's already in base e.

Use the 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.

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 Log_{10}(x), and Natural Logarithm is defined as Log_{e}(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 fewer 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.

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”.