The natural or neperian logarithm is said to have base e because ln(e) = 1. Furthermore, the neperian logarithm of a number x can also be defined as the power to which e must be raised in order to obtain x. Furthermore, the neperian logarithm function is thus the reciprocal bijection of the exponential function.

We call the neperian logarithm of a strictly positive real a, the unique solution of the equation ex = a

## I. Neperian logarithm function, reciprocal function of the exponential function.

Properties: the exponential function.

The exponential function is continuous and strictly increasing on .
We have and .

The equation , with , then has a unique solution in , according to the intermediate value theorem.

Definition: neperian logarithm function.

The function defined on which associates to any strictly positive real number x the unique solution of the equation with unknown value is called the neperian logarithm function, noted .

We define as follows.

Example:

Using the key on the calculator, we can check that .

Remark:

When there is no ambiguity, we can note instead of .

Properties: neperian logarithm function.
• For any real
• For any real

Example:

and .

## II. Curves of the neperian logarithm and exponential functions

Ownership:

In an orthonormal frame of reference, the representative curves of the functions and are symmetrical with respect to the line of equation y= x.

## III. Direction of variation of the neperian logarithm function

Ownership:

The function is strictly increasing on .

Demonstration:

Let a and b be two strictly positive real numbers.
.

We deduce because the function is strictly increasing on .

Properties:

For all real a > O and b > O : .

Proof:

because the function is strictly increasing on .
because the function is strictly increasing on .

Remark:

and .

## IV. Algebraic properties of the neperian logarithm function

### 1.functional relationship.

Ownership:

For all real a and b strictly positive :
.

Proof:

For all real a and b strictly positive,
or .
So we have .

Remarks:

1. We find the particularity that this function transforms the products into sums.
2. This formula generalizes to a product of several factors.

Examples:

### 2. Logarithm of an inverse and a quotient.

Ownership:

For all real a and b strictly positive :

and .

Proof:

For any real number a strictly positive :

hence

so we have .

For all real numbers a and b strictly positive:

.

### 3. Logarithm of a power, of a square root.

Ownership:

For any real a strictly positive, and for any relative integer n :

and .

Examples:

.

.

## V. Study of the neperian logarithm function

### 1.derivative of the neperian logarithm function.

Ownership:

The function is derivable on and, for any real , .

Proof:
We admit that the function is derivable on .

For any real , we pose .
The function being derivable on and the exponential function being derivable on ,

f is also derivable on as a composite of derivable functions.
Knowing that , posing and , we then have :
.
We also have so .
Therefore, we have .

Properties:

and

Ownership:

Ownership:

### 5. Compound function ln (u).

Property: derivative of ln u.

Let u be a strictly positive differentiable function on an interval l.
The function is then derivable on I and

Property : direction of variation of ln(u).

Let u be a strictly positive differentiable function on an interval l.
The functions u and have the same direction of variation on l.

Proof:

u being strictly positive, the sign of is the same as that of .

Or , which means that the sign of is the same as that of ,

i.e. u and have the same direction of variation.

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