When looking at natural logarithms, there are three values that can be calculated. These three values are the ln 0, ln 1, and ln n, where n is any positive integer.

The ln 0 is a special case where n is zero. This case can be easily proven by * using simple algebra*.

The ln 1 is called the inverse of the natural logarithm of one. This case can also be easily proven by using simple algebra.

The *ln n expressed* as a **single natural logarithm** is a little more complex to prove, but can be done so nonetheless. This article will go into detail about how to prove this statement.

## Next, find the single natural logarithm of 3

Now, find the ** single natural logarithm** of 9. As with the last step, you can either calculate this yourself or look it up online. Calculating it yourself is more satisfying and teaches you something new!

To calculate the single natural logarithm of 9, first subtract 9 from 10 and then divide by 2.5. This gives you an answer of . Then take the inverse of this number, or . Subtract 1 from this number and you have your answer:

This is very interesting! We know that 3ln3=ln9, but we have found that ln9 can be expressed as a combination of ln3 and ln(-3)!.

In mathematics, there are many types of equations that **require different solutions**. One type of equation is solving for a variable within a complex equation.

## Finally, combine these two values together

So, to find the natural logarithm of a sum or difference of natural logariths, subtract the smaller value from the larger and then find the natural logarithm of that difference. Then, add that value to the last found N ln value and you have your answer!

For example, let’s find the natural logarithm of *3 ln 3 − ln 9 expressed* as a single natural logarithm.

We start by finding the difference between 3 ln 3 and ln 9: **ln 9 − 3 ln 3** . We then find the natural logarithm of that difference: 1ln 9 . Next, we add 1ln 9 to 1ln 2 (the last found N ln) to get 2ln 2 .

Therefore, the answer is: 2.

## The answer is ln(9)/ln(3) = 2.36719266667

In this article, we explored the concept of *3 ln 3* –

**ln 9 expressed**as a single natural logarithm. We saw how to express this equation as a single logarithm and explored some interesting properties of the expression.

We began by rewriting the expression as:

Then we factored out the natural logarithm of 9, leaving us with an equivalent expression of 3 ln 3 – ln 9 in terms of natural logs. We then simplified this expression and did some rearranging to get our final answer: 2log(3) – 2log(9).

As this article has shown, the concept of 3 ln 3 – ln 9 expressed as a single natural logarithm is a tricky one.

## This result can also be expressed as e^{2.36719266667}

In this expression, e is the base of the natural logarithm ln and 2.36719266667 is the third power of e. The number preceding the logarithm determines which power of the base you are raising a number to.

As with all logarithms, the ln 9 cannot be calculated by direct multiplication or division, but only by applying an inverse function, in this case, an inverse logarithm.

The result can also be expressed as **ln 9 ≈ 2**.36719266667·10^{−2}. This can be read as “nine equals two point three six seven nine two six six seven two of ten to the minus two”. The last part of that sentence is simply reading out the number that would correspond to one unit in the natural logarithm dimension.

Logarithms come in different bases other than e and 10, but these are not as commonly used.

## What does this mean?

When the ** natural logarithm** of a number is expressed as a single logarithm, it means that the number is in the nth row of the

**natural logarithm table**.

For example, 1 is in the first row, so when you take the natural logarithm of 1, you *get ln 1*, or 0. The second number in the first row is 2, so taking the natural logarihm of 2 gives you ln 2 or 0.7.

The third number in the first row is 3, so taking the natural logarithm of 3 gives you ln 3 or 0.47. All of these numbers are already in their own rows, which makes it easy to tell which ones they are.

The same goes for all of the rows after the first one.

## How do I remember this?

The easiest way to remember this is by thinking about how we write numbers.

We write all numbers as a sum of powers of ten. One is a tenth, two is a hundred, three is a thousand, and so on. We call these units tenths, hundreds, thousands, and so on up to tens of thousands.

So, how do we * express one natural logarithm* as a

**single logarithm**? Well, we can only express one natural logarithm as a single logarithm if we express it as a single unit: one thousand.

One thousand is the same number of units in one natural logarithm.

## Is there an easier way to calculate this?

Yes! There is a way to calculate what is known as the natural logarithm of any number from 3 to 9. The trick is to use the ln 9 and **ln 8 numbers**, which are called intergers.

The natural logarithm of 3 is 1, **since ln 3** = 1. To find the natural logarithm of 5, take the natural logarithm of 4 (which is 2) and add 1.

Since we are dealing with numbers that are less than 10, you can approximate the multiplication as just adding 1. So the natural logarithm of 5 is 2+.1=2.1.

The same method can be applied to 6 and 7, and 8 and 9 respectively. Given enough time, you will be able to recognize patterns in these natural logarithms.

## What is a natural logarithm?

A natural logarithm is the inverse function of an exponential function. For example, if y is the expression for x raised to a certain power (exponential), then x is the expression for y divided by a certain power (natural logarithm).

Logarithms are very useful in engineering applications, so it’s important to understand how to convert between natural and common logarithms. The rest of this article will discuss how to do this, so read on!

As mentioned above, there are **three common logarithms** between 3 and 9, with no single one being the average. However, there is a **single natural logarithm** between these two numbers: 5. No other values between 3 and 9 have a natural logariyh of 5, which makes it the average of all **three common logs**.