Neperian logarithm: senior math course download PDF.

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 \mathbb{R}.
We have \lim_{x\to\,-\infty\,}e^x=0 and \lim_{x\to\,+\infty\,}e^x=+\infty.

The equation e^x=k, with k\in\,\mathbb{R}^{+*}, then has a unique solution in \mathbb{R}, according to the intermediate value theorem.

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Definition: neperian logarithm function.

The function defined on %5D0;+\infty%5B which associates to any strictly positive real number x the unique solution of the equation e^y=x with unknown value y is called the neperian logarithm function, noted ln.

We define y\,=\,ln\,(x) as follows.

Example:

Using the ln key on the calculator, we can check that ln\,(2)\,\simeq\,0,693.

Remark:

When there is no ambiguity, we can notelnx instead of ln(x).

Properties: neperian logarithm function.
  • For any real x>0,e^{lnx}=x
  • For any real x,ln(e^x)=x
  • ln1=0
  • lne=1
  • ln(\frac{1}{e})=-lne=-1

Example:

ln(e^3)=3 and e^{ln3}=3.

II. Curves of the neperian logarithm and exponential functions

Ownership:

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

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III. Direction of variation of the neperian logarithm function

Ownership:

The function ln is strictly increasing on %5D0;+\infty%5B.

Demonstration:

Let a and b be two strictly positive real numbers.
0<a<b\Leftrightarrow\,0<e^{ln(a)}<e^{ln(b)}.

We deduce ln\,(a)\,<\,ln\,(b) because the function x\,\mapsto  \,e^x is strictly increasing on \mathbb{R}.

Properties:

For all real a > O and b > O : ln(a)=\,ln(b)\,\Leftrightarrow\,a=b\,\,et\,ln\,(a)\,<\,ln\,(b)\,\Leftrightarrow\,a\,<\,b.

Proof:

ln(a)=ln(b)\Leftrightarrow\,e^{ln(a)}=e^{ln(b)}\Leftrightarrow\,a=b because the function x\,\mapsto  \,e^x is strictly increasing on \mathbb{R}.
ln(a)<ln(b)\Leftrightarrow\,e^{ln(a)}<e^{ln(b)}\Leftrightarrow\,a<b because the function x\,\mapsto  \,e^x is strictly increasing on \mathbb{R}.

Remark:

ln(x)>0\Leftrightarrow\,x>1 and ln(x)<0\Leftrightarrow\,0<x<1.

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IV. Algebraic properties of the neperian logarithm function

1.functional relationship.

Ownership:

For all real a and b strictly positive :
ln(ab)\,=\,ln(a)\,+\,ln(b).

Proof:

For all real a and b strictly positive, e^{ln(ab)}=ab=e^{ln(a)}\times  \,e^{ln(b)}=e^{ln(a)+ln(b)}
or .
So we have ln(ab)\,=\,ln(a)\,+\,ln(b).

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:

ln\,(10)\,=\,ln\,(5\,\times  \,2)\,=\,ln\,(5)\,+\,ln(2)
ln\,(30)\,=\,ln\,(2\times  \,3\,\times  \,5)\,=ln\,(2)\,+\,ln\,(3)\,+\,ln\,(5)

2. Logarithm of an inverse and a quotient.

Ownership:

For all real a and b strictly positive :

ln(\frac{a}{b})=ln(a)-ln(b) and ln(\frac{1}{a})=-ln(a).

Proof:

For any real number a strictly positive :

ln(1)=ln(\frac{a}{a})=ln(a\times  \,\frac{1}{a})=lna+ln(\frac{1}{a}) hence ln(a)+ln(\frac{1}{a})=0

so we have ln(\frac{1}{a})=-ln(a).

For all real numbers a and b strictly positive:

ln(\frac{a}{b})=ln(a\times  \,\frac{1}{b})=ln(a)+ln(\frac{1}{b})=ln(a)-ln(b).

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

Ownership:

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

ln(a^n)\,=\,nln(a) and ln(\sqrt{a})=\frac{1}{2}ln(a).

Examples:

ln(25)=ln(5^2)=2ln(5).

ln(16)-2ln(2)+ln(8)=ln(2^4)-2ln(2)+ln(2^3)=4ln(2)-2ln(2)+3ln(2)=5ln(2)

ln(\sqrt{6})=ln(6^{\frac{1}{2}})=\frac{1}{2}ln(6).

V. Study of the neperian logarithm function

1.derivative of the neperian logarithm function.

Ownership:

The function ln is derivable on %5D0;+\infty%5B and, for any real x>0, ln'(x)=\frac{1}{x}.

Proof:
We admit that the function ln is derivable on %5D0;+\infty%5B.

For any real x>0, we pose f(x)=e^{ln(x)}.
The function ln being derivable on %5D0;+\infty%5B and the exponential function being derivable on \mathbb{R},

f is also derivable on %5D0;+\infty%5B as a composite of derivable functions.
Knowing that (vou)'=(v'ou)\times  \,u', posing v(x)=e^x and u(x)\,=\,ln(x), we then have :
f'(x)=e^{ln(x)}\times  \,ln'(x)=x\times  \,ln'(x).
We also have f(x)=x so f'(x)=1.
Therefore, we have x\times  \,ln'(x)=1\Leftrightarrow\,ln'(x)=\frac{1}{x}.

2.limits to the bounds of the definition set.

Properties:

\lim_{x\to\,+\infty}ln(x)=+\infty and \lim_{x\to\,0}ln(x)=-\infty

3. table of variations of ln and representative curve.

Ownership:

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4.comparative growth.

Ownership:

\lim_{x\to\,+\infty}\frac{lnx}{x}=0\,;\,\,\lim_{x\to\,0}xlnx=0;\lim_{x\to\,+\infty}\frac{lnx}{x^n}=0\,(n\in\mathbb{N}^*)\,;\,\,\lim_{x\to\,0}x^nlnx=0\,(n\in\mathbb{N}^*)

5. Compound function ln (u).

Property: derivative of ln u.

Let u be a strictly positive differentiable function on an interval l.
The function lnu is then derivable on I and (ln\,u)'\,=\frac{u'}{u}

Property : direction of variation of ln(u).

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

Proof:

u being strictly positive, the sign of \frac{u'}{u} is the same as that of u'u'.

Or (ln\,u)'\,=\frac{u'}{u}, which means that the sign of (lnu)' is the same as that of u',

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

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