Contents

- Definition
- Concept of the Number “e”
- Graphical Analysis of the Exponential function
- Graphical Analysis of the Natural Logarithmic Function
- Relationship between Exponential and the Natural Logarithmic Function
- Properties of Natural Logarithm
- Differentiation of Natural Logarithmic Function
- Integration of Natural Logarithmic Function
- Examples

# Natural Logarithm|Definition & Meaning

## Definition

The natural logarithm notated “ln” is the **logarithm** of a number to the base “**e,**” which is a mathematical **constant** known as **Euler’s** number. It is approximately equal to **2.718281828,** which is a transcendental and **irrational** number. The natural logarithm is commonly abbreviated as “**ln.**”

**Figure 1** shows the demonstration of the **natural logarithm** of **x**.

The **natural** logarithm of a number **x** defines how many times “**e**” is used in **multiplication** to get the given number **x**. It is the **power** of **e** to which it would be raised to become **equal** to the number **x**.

For **example**, the natural **logarithm** of 8.5 is **2.14006**. This means that the number **e** is multiplied** 2.14006** times to get **8.5**. So, both can be written as:

ln(8.5) = 2.14006

e$^{2.14006}$ = 8.5

**Logarithms** can have any **positive** base other than** 1,** which will be its constant **multiplier**. In **natural** logarithms, the constant multiplier is **e,** as its **base** is **e**.

## Concept of the Number “e”

To understand the **natural** logarithm, one must understand its **base** “**e**.” It is an **irrational** number as it is **non-recurring** and non-terminating. It can be written to a million **decimal** places, and so on, but it won’t have a repetitive pattern.

It was used by a **scientist** named **Leonhard Euler,** who derived its concept based on the different **exponential** growths happening in the universe.

Jacob **Bernoulli** discovered the constant** e** and formed the following **equation**:

\[ \lim_{ n \to \infty } { \Big( 1 + \frac{1}{n} \Big) }^{n} = e \]

The concept of “**e**” also comes from **compound interest,** as it also refers to the exponential growth of **money**.

## Graphical Analysis of the Exponential function

An exponential **function** has the variable **x** as the **exponent**. Mathematically, it is:

f(x) = b$^x$

Where **b** is a **positive** number except** 1**.

The most common **exponential** function is the **function** with the base “**e.**”

**Figure 2** shows the **graph** for the exponential function:

f(x) = e$^x$

The **graph** starts from **negative infinity** and goes to **positive** infinity. Point** A** represents the coordinates **(0, 1),** and point **B** as **(1, e)**.

This **exponential** function is a very **special** function in mathematics because of its following **properties**.

### Slope

The **graph** of the **exponential** function y = e$^x$ is a curve. It has a different slope at each **point** on the curve. If a **tangent** is drawn at a point, then its **slope** will be the slope of the **curve** at that point.

If a **tangent** line is drawn at **point B**, then its slope comes out to be **e** which is the **same** as the **value** of the function** f(x)** at that point. Also, at point **A**, the **slope** is **1** same as the function’s value.

Turns out that the **slope** at each point on the **curve** is equal to the **value** of the **function** at that **point**. This unique behavior makes it a **special** function.

Hence, it can be stated that the **slope** of the **exponential** function y = e$^x$ at a **point** on the **curve** is equal to the **value** of the curve at that point.

### Area

Another unique trait is the **area** under the curve of the **exponential** function. Let **L** be any **point** on the curve. The area under the curve from **negative infinity** to the point **L** on the curve equals the **value** of the **function** at **L**.

No such **function** has this behavior in terms of its **area** and **slope**. That is why it is considered a **unique** function.

## Graphical Analysis of the Natural Logarithmic Function

**Figure 3** shows the graph for the function:

f(x) = ln(x)

Point **P** has the coordinates **(1,0)** and point **Q** has the coordinates **(e,1)**. The unique **properties** of an exponential function make the **logarithmic** function also special. The base number “**e**” is common in both functions.

## Relationship between Exponential and the Natural Logarithmic Function

The function **ln(x)** is the **inverse** of the function **e$^x$**, provided that the **function** ln(x) is a **real-valued** function of a **positive** variable. From this concept, the following **identities** are formed:

e$^{\ln (x)}$ = x

ln(e$^{x}$) = x

**Figure 4** shows the **exponential** and the **natural logarithmic** function.

The function **ln(x)** is graphed by **flipping** the **e$^x$** function over the line** x = y**.

Natural logarithms are **used** to solve **equations** in which the unknown **variable** is equal to an **exponent** of some other value. The problems of the **decay** constant and **half-life** are solved by using **natural** logarithms as they evolve **exponentially**.

**Logarithms** are used in solving problems related to **compound** interest.

## Properties of Natural Logarithm

The following are the **important** properties of the **natural** logarithm:

ln(e) = 1

This is because e$^1$ = e.

ln(1) = 0

This is because e$^0$ = 1**.**

The **multiplication** of positive numbers **x** and **y** maps into **addition** in natural logarithms as:

ln(xy) = ln(x) + ln(y)

As for **division**:

ln(x/y) = ln(x) – ln(y)

The **power** **y** inside the** ln** function can be **multiplied** with the natural logarithm as:

ln (x$^{y}$) = y * ln(x)

Also, as **x** approaches the **limit 0**, the **fraction** ln(1 + x) / x becomes **1** as:

\[ \lim_{ x \to 0 } {\frac{\ln(1 + x)}{x} } = 1 \]

## Differentiation of Natural Logarithmic Function

The **differential** of the function **ln(x)** is given as follows:

\[ \frac{d}{dx} \ln(x) = \frac{1}{x} \]

The **differentiation** of the function **ln(ax)**, where** a** is a constant, is:

\[ \frac{d}{dx} \ln(ax) = \frac{d}{dx} [\ln(a) + \ln(x)] = \frac{d}{dx} \ln(a) + \frac{d}{dx} \ln(x)= \frac{1}{x} \]

Hence, the **constant** does not **change** the differential.

## Integration of Natural Logarithmic Function

The **integral** of the function ln(x) is as follows:

\[ \int \ln(x) \ dx = x\ln(x) \ – \ x + c \]

Where **c** is a **constant** of integration.

## Examples

### Example 1

Solve e$^{7\ln(y)}$.

### Solution

As:

7 * ln(y) = ln(y$^7$)

So:

e$^{7\ln(y)}$ = e$^{\ln(y^7)}$

The** e** and** ln** cancel to give the answer as:

**e$^{7\ln(y)}$ = y$^7$**

### Example 2

Solve the **equation:**

2 * ln(x) – 3 * ln(1/x) = 10

### Solution

As:

1 / x = x$^{-1}$

So:

2 * ln(x) – 3 * ln(x$^{-1}$) = 10

2 * ln(x) – 3(-1)ln(x) = 10

2 * ln(x) + 3 * ln(x) = 10

5 * ln(x) = 10

Dividing **5** on both sides gives:

ln(x) = 2

So, the final **exact** value is:

**x = e$^2$**

*All the images are created using GeoGebra.*