# Log Vs. In: Differences with Examples

A natural logarithm is a power to which the base ‘e’ must be raised to acquire a number known as its log number. The exponential function is denoted by e. John Napier, who discovered and developed the notion of logarithms, was the first to discover it in the 17th century.

Before delving into the fundamental differences between Log and Ln, let’s first define log and ln through this article.

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**Definition of Log**

The logarithm to the base 10, i.e., natural logarithm, is defined in mathematics as the inverse function of exponentiation.

- In simplified terms, the logarithm is the power to which a number must be increased to get any other number.

- It is also known as the base-10 logarithm or the common logarithm.

- A logarithm’s general form is denoted as logₐ (y) = x

Which can also be written as ax = y

**Properties of logarithm **

The four fundamental features of logarithm are listed here to assist you in solving logarithmic issues.

**1- Logb(mn) = Logb m plus Logb n**

This logarithmic characteristic states that multiplying two logarithm values is equal to adding the separate logarithms, respectively.

**2- Logb (m/n) = Logb m minus Logb n**

This logarithmic property states that dividing two logarithm values is equal to subtracting the individual logarithm respectively.

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**3- n logbm = logb (mn)**

The above condition is known as the logarithmic, exponential rule. The logarithm of m plus the rational exponent equals the exponent multiplied by its logarithm, respectively.

**4- Logb m is equal to loga m / loga**

When two numbers with the same base are split, the exponents are subtracted, respectively.

**Definition of Ln in simple terms**

The natural logarithm is abbreviated or written as Ln. It is called the logarithm of base e. In this case, e is an irrational and transcendental number that is roughly equivalent to 2.718281828459… The natural logarithm (ln) is denoted by the symbol ln x or loge x.

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**What Are the Primary Distinctions or differences Between Log and Ln?**

To answer logarithmic issues, one needs to understand the difference between log and ln. Grasp distinct topics may be aided by having a basic understanding of the logarithm function. The following are some of the critical distinctions between natural logarithm and logarithm:

**Log: **

- A logarithm to the base 10 is referred to as a log.

- This is sometimes referred to as a common logarithm.

- The standard log is denoted by log10 (x)

- The common logarithm has the exponent form 10x =y.

- The common logarithm’s interrogative sentence is “At what number should we increase 10 to obtain y?”

- When compared to ln, it is more extensively employed in the subject of physics.

- Log base 10 is how it is expressed mathematically.

**Ln: **

- Ln is a logarithm to the base e.

- This is also referred to as a natural logarithm.

- The natural log is denoted by loge (x)

- The natural logarithm’s exponent form is ex = y

- The natural logarithm’s interrogative phrase is “At what number should we increase Euler’s constant number to obtain y?”

- Because logarithms are generally followed to their logical conclusion in physics, ln is employed considerably less often.

- This is expressed mathematically as log base e.

Let us go through several rules of the Ln in this article.

**1- first rule: Quotient Rule**

- ln(x/y) equals ln(x) – ln(y)
- The difference between the ln of x and ln of y is equal to the natural log of the division of x and y.
- For ex, ln(10/5) = ln(10) – ln (5)

**2- second rule: Power Rule.**

- ln(XY) is the same as y * ln (x)
- The natural log of x multiplied by y is y times the ln of x.
- For ex, ln(42) is 2 * ln (4)

**3- third rule: Reciprocal Rule.**

- ln(1/x) is the same as ln (x)
- The natural log of x’s reciprocal is equal to the inverse of x’s ln.
- For instance, ln (13) = -ln (3)

**What Is the Difference Between a Log and an ln Graph?**

Let’s look at some of the essential distinctions between Log and Ln:

To answer logarithmic issues, one must first understand the distinction between log and natural log. A solid understanding of exponential functions may aid distinct grasp topics.

Example to make things more clear.

In science, logarithmic scales are used: Logarithmic scales appear in a surprising variety of scientific and everyday events because logarithms connect multiplicative changes to incremental increases. Consider the following example of sound intensity:

To enhance the loudness of a speaker by 10 decibels (dB), the power must be multiplied by ten. Similarly, 20 dB necessitates 100 times the power, whereas 30 dB necessitates 1,000 times.

Decibels are said to “advance arithmetically” or “variate on a logarithmic scale” because they fluctuate proportionately with the logarithm of another measurement, in this instance, the strength of the sound wave, which “progresses geometrically” or “varies on a linear scale.”

**Questions as an example: **

**1) Solve for y in log2 y = 3**

**Solution:** The above function’s logarithm function may be expressed as 2^3 = y.

As a result, 2^3 = 2 x 2 x 2 = 8 or y = 8.

**2) Simplify the following expression ie log (96).**

**Answer:** We shall apply the Log and ln principles that we explained before. Since we know that the number 96 is not an excellent clean power of 10 (the way that 100 was), we cannot be creative with exponents to arrive at an accurate solution.

So we can calculate the number by entering this into a calculator and remembering to use the “LOG” key (not the “LN” option), and we obtain log(96) = 1.98227123304…, or log(96) = 1.98, rounded to two decimal places.

**3) Evaluate the following expression log 3 150 – log 3 2**

**Solution:** Here, we apply the quotient rule law, as the subtraction is evolved

log 3 486– log 3 2 = log 3 (486/2)

= log 3 243

Write the argument in exponential form

243= 3 5

Hence, the answer is 5.

**Ending with an interesting fact **

Michael Stifel, a German mathematician, and thinker, was the first to introduce the notion of a logarithm into contemporary times (around the year from 1487 to 1567).

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Logarithm with base 10 is called a common logarithm or a Briggsian logarithm, and it may also be represented as log n. They are frequently written without foundation.

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