# 4 Natural Log Rules You Must Know

February 5, 2024

Chances are that if you are taking high school or college math classes, at some point you’ll need to understand natural logarithms, commonly known as *natural logs* (written as *ln*)*. *This concept, often taught in 11th and 12th-grade math, is tricky for many students. Especially if you plan to continue studying math after high school, or if you plan to study in a field with a good amount of applied math such as economics, you will certainly want to familiarize yourself with the concept of natural log rules. On the other hand, even if you don’t consider yourself to be a “math person,” or if you never want to see another math problem after high school, do not fret! We’re going to explain this concept to you step by step.

First, we’ll go over logarithms in general, followed by the definition of *e *(an essential concept to understand for natural logs). Then we’ll go over the difference between *logs *and *natural logs*. Lastly, we’ll go over some important rules and properties, with a few brief examples. Once you understand these definitions and key rules of natural logs, we hope that you’ll be better equipped to reason your way through those problems thrown at you on the next exam.

**Here’s the general idea**

A logarithm is the opposite of a power, which means that when we take a logarithm of a number, we undo an exponentiation. For example, if we are to raise *x* = 5 to the power of *a* = 3, the expression looks like x* ^{a}* = 5

^{3}. The result, due to the basic rules of exponentiation, is basically the same as multiplying 5 * 5 * 5. The answer is 125.

The equation looks like: *x ^{a} *= 5

^{3}= 125

Let’s say we have an equation in which we don’t know that *a* = 3. So now this equation looks like, 5* ^{x} *= 125. This is where the logarithmic function comes in. We would say “log base 5 of 125” and then find the answer for

*a*.

This equation looks like: **log _{5}125 = a = 3**

*Natural Log Rules (Continued)*

Of course, we already knew that the answer to this problem would be 3, since we probably have it memorized from when we learned simple cubes. But when the problem becomes more complicated, or if it is not one of the cubes you have memorized, the logarithm becomes a necessary tool.

Now for the next step: the *natural log* (AKA *ln*). The natural log is the inverse of *e. *So, *e ^{x }*and

*ln*(

*x*) are inverses.

Put differently: we know that a number cubed and the cube root of that number are inverses. In the same way that taking the cube root of *x ^{3} *equals

*x*,

*ln*(

*e*) simplifies to

^{x}*x*.

__Before we go any further, what exactly is __*e*?

__Before we go any further, what exactly is__

*e*?The letter *e *represents a natural constant. A natural constant is, in other words, a number that arises by a principle in nature, which will always remain the same in every scenario (*pi*, or π, or 3.14159 etc etc etc, is another or example of a natural constant, which you’ve likely already seen in math class).

In this case, *e *is known as the *natural exponent*, with a value of approximately **2.71828**.

*e *appears in a number of common instances in mathematics, such as in growth equations (for processes increasing quantity over time at increasing rates) and decay equations (for processes of reducing quantity over time at reducing rates). For example, when it comes to growth equations *ln(x) *is the time needed to grow *x*, while *e*^{x }is the amount of growth that has occurred after time *x*.

__Now, back to natural logarithms…__

__Now, back to natural logarithms…__

So how does all of this help us to solve equations?

Since *e *is so commonly found in math (as well as fields involving applied math, such as economics), the *natural log* was created as a shortcut for calculating a logarithm with a base of* e*. Written as an equation, **ln( x) = log_{e}(x)**. In other words, the

*ln*lets people know that an equation is taking the logarithm, with a base of

*e*, of a number.

Here’s a simple example: to solve for *e ^{x} = *7, you can solve for

*x*by taking the natural log of both sides of the equation:

*e ^{x} *= 7

*ln(e ^{x}) *=

*ln*(7)

*x* = *ln*(5)

Here, ln(*e ^{x}*) simplifies to

*x*, similar to how taking the cube root of

*x*

^{3}simplifies to

*x*. Continue reading for more information on natural logs versus other logs, as well as main rules and properties of natural logs.

**What’s the difference between natural logs and other logarithms?**

As stated above, a logarithm is the opposite of a power, in other words, the undoing of an exponentiation. Because a natural logarithm is a kind of logarithm, this undoing applies to natural logs as well.

But what’s the difference between a natural log and any other log? This is a common question when learning about these topics. The main difference between natural logs and other logs is the *base* being used. In most logarithms, any base can be used (often, the base is 10). In natural logarithms, on the other hand, the base will always be *e*. This means, *ln*(*x*) = **log_{e}(x)**.

**Four main rules of natural logs**

Below are the four rules you will come across when working with natural logs. You will be sure to come across these rules when working with logarithm equations, so make sure to review them.

Since taking the logarithm is essentially the opposite of taking an exponentiation, the rules of logarithms are similar to the rules of exponents.

__1) Product Rule__

__1) Product Rule__

When it comes to exponents, the rule goes like this: **𝑥**^{a}**x ^{b}**

**=**

**x**. In other words, when the equation asks you to multiply two values raised to powers, you should add the two powers to solve the problem.

^{a+b}For natural logarithms, it becomes: **ln(x)(y) = ln(x) + ln(y)**

In other words, the natural log of the multiplication of *x* and *y* is the sum of the *ln *of *x *and the *ln *of *y.*

Here’s an example with some numbers: *ln*(5)(3) = *ln*(5) + *ln*(3)

__2) Quotient Rule__

__2) Quotient Rule__

For exponential division problems, on the other hand, the rule goes: **𝑥**^{a }**/ x ^{b}**

**=**

**x**. In other words, when the equation asks you to divide two values raised to powers, you should subtract the two powers to solve the problem.

^{a-b}With natural log equations, then, the rule becomes: **ln(x/y) = ln(x) – ln(y)**

In other words, the natural log of the division, *x* divided by *y,* is the difference of *ln *of *x* and the *ln *of *y.*

Here’s an example with some numbers: *ln*(5/3) = *ln*(5) – *ln*(3)

*Natural Log Rules (Continued)*

__3) Reciprocal Rule__

__3) Reciprocal Rule__

For exponential problems, the rule goes: **x ^{-a} = 1/x^{a}**. In other words, when a value is raised to a negative power, it is equal to dividing the number one by the value to its positive power.

With natural log equations, the rule states: **ln(1/x) = -ln(x)**

To state simply, the natural log of the reciprocal of *x* is the negative value of the *ln *of *x.*

Here’s an example with some numbers: *ln*(1/5) = –*ln*(5)

__4) Power Rule__

__4) Power Rule__

For exponential equations, the power rule goes, **(x ^{a})^{b} = x^{ab}**. In other words, when a power is raised to a power

*a*and then a power

*b*, it is the same as raising the number to the power

*a*times the power

*b*.

With natural log equations, the rule then becomes: **ln(x ^{y}) = y * ln(x)**

In other words, the natural log of *x* to the power of *y *is *y *times the *ln* of *x*.

Here’s an example with some numbers: *ln*(5^{3}) = 3 * *ln*(5)

**Properties you should know**

In addition to the four main rules described above, there are several properties it will be useful to understand specifically when using natural logarithms. These situations may come up on exams, so memorizing them now can help you to work with greater efficiency later.

In the case of taking the *ln *of a negative number: *ln *of a negative number is undefined.

In the case of taking the *ln *of 0: *ln *of 0 is undefined.

Next, in the case of taking the *ln *of infinity: *ln*(**¥****) = ****¥**

In the case of taking the *ln *of *e *raised to the *x *power: *ln*(*e ^{x}*) = x

In the case of taking *e *raised to the *ln* power: *e ^{ln(x)} *=

*x*

*Natural Logarithm Rules (Continued)*

In the case of taking the *ln* of 1: ** ln(1) = 0**. This is because the formula for the log of 1 comes from the formula for the power of 0. In other words,

*e*

^{0}= 1, and therefore ln(0) = 1.

In the case of taking the *ln *of *e*, ** ln(e) = 1**. This is because the formula for the log of

*e*is the same as the formula for the power of 1,

*e*=

^{1}*e.*Therefore,

*ln*(

*e*) = 1.

**Natural Log Rules Summary**

In many cases in math or other courses, you will be using the natural log, or *ln*, which is the inverse of *e*. The main difference between natural logs and other logarithms is that in natural logs, the base is always *e*, which represents a natural constant with a value of approximately 2.71828. Though the topic of natural logarithms poses challenges for many students, you can best tackle the problems at hand by first understanding the four main rules of natural logs, as well as a number of other simple properties.

## Natural Logarithm Rules – More on math topics…

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