1 Definition and Properties of the Natural Log Function









Properties of Exponents and Logarithms

Properties of Logarithms (Recall that logs are only defined for positive values of x.) For the natural logarithm For logarithms base a. 1. lnxy = lnx + lny. 1.
Exponents and Logarithms


6.2 Properties of Logarithms

(Inverse Properties of Exponential and Log Functions) Let b > 0 b = 1. We have a power
S&Z . & .


FONCTION LOGARITHME NEPERIEN

exp et ln sont symétriques par rapport à la droite d'équation y = x. - Dans le domaine scientifique on utilise la fonction logarithme décimale
LogTS






Elementary Functions Rules for logarithms Exponential Functions

+ 4). By the first inverse property since ln() stands for the logarithm base e
. Working With Logarithms (slides to )


Logarithmic Functions

Natural Logarithmic Properties. 1. Product—ln(xy)=lnx+lny. 2. Quotient—ln(x/y)=lnx-lny. 3. Power—lnx y. =ylnx. Change of Base. Base b logax=logbx.
LogarithmicFunctions AVoigt


11.4 Properties of Logarithms

and turn them into adding subtracting or coefficients on the outside of the logarithm


Limits involving ln(x)

Using the rules of logarithms we see that ln 2m = m ln 2 > m/2
. Limits Derivatives and Integrals





LOGARITHME NEPERIEN

.. x ∈ IR+. * y = ln x. ⇔ y ∈ IR e y. = x traduit le fait que les fonctions exponentielle et logarithme népérien sont réciproques l'une ...
ln


Physics 116A Winter 2011 - The complex logarithm exponential and

Consider the logarithm of a positive real number. This function satisfies a number of properties: eln x = x. (17) ln(ea) = a
clog


213212 1 Definition and Properties of the Natural Log Function Lecture 2Section 7.2 The Logarithm Function, Part I

Jiwen He

1 Definition and Properties of the Natural Log

Function

1.1 Definition of the Natural Log Function

What We Do/Don"t Know Aboutf(x) =xr?

We know that:•Forr=npositive integer,f(x) =xn=ntimes???? x·x···x. To calculate 26, we do in our head or on a paper

2×2×2×2×2×2,

but what does the computer actually do when we type

2^6•Forr= 0,f(x) =x0= 1.•Forr=-n,f(x) =?1x

n,x?= 0.?x-1=1x .•Forr=pq rational,f(x) =y,x >0, whereyq=xp.f(x) =x1n is the inverse function ofg(x) =xnforx >0.?g◦f(x) =? x1n n=x.•Properties (randsrational) x r+s=xr·xs, xr·s=?xr?s, ddx xr=rxr-1,? x rdx=1r+ 1xr+1+C, r?=-1.

We DO NOT knowyetthat:

x -1dx=?1x dx=? andxr=? forrreal.1

What is the Natural Log Function?

Definition 1.The function

lnx=? x 11t dt, x >0, is called thenatural logarithm function.•ln1 = 0. •lnx <0 for 0< x <1, lnx >0 forx >1.•d dx (lnx) =1x >0?lnxis increasing.•d 2dx

2(lnx) =-1x

2<0?lnxis concave down.

1.2 Examples

Example 1:lnx= 0and(lnx)?= 1atx= 1Exercise 7.2.23

Show that

lim x→1lnxx-1= 1.Proof. lim x→1lnxx-1= limx→1lnx-ln1x-1=ddx (lnx)????x=1=1x ???x=1= 1.The limit has theindeterminate form?00 ?and is interpreted here in terms of thederivativeof lnx.2

Example 2:lnxandx-1Exercise 7.2.24(a)

Show thatx-1x

lnx=? x 11t dt=1c (x-1).•Ifx >1, then1x <1c <1 andx-1>0 so (1) holds.•If 0< x <1, then 1<1c <1x andx-1<0 so (1) holds.Example 3:lnnand Harmonic NumberExercise 7.2.25(a)

Show that forn≥2

12 +13 +···+1n Jiwen He

1 Definition and Properties of the Natural Log

Function

1.1 Definition of the Natural Log Function

What We Do/Don"t Know Aboutf(x) =xr?

We know that:•Forr=npositive integer,f(x) =xn=ntimes???? x·x···x. To calculate 26, we do in our head or on a paper

2×2×2×2×2×2,

but what does the computer actually do when we type

2^6•Forr= 0,f(x) =x0= 1.•Forr=-n,f(x) =?1x

n,x?= 0.?x-1=1x .•Forr=pq rational,f(x) =y,x >0, whereyq=xp.f(x) =x1n is the inverse function ofg(x) =xnforx >0.?g◦f(x) =? x1n n=x.•Properties (randsrational) x r+s=xr·xs, xr·s=?xr?s, ddx xr=rxr-1,? x rdx=1r+ 1xr+1+C, r?=-1.

We DO NOT knowyetthat:

x -1dx=?1x dx=? andxr=? forrreal.1

What is the Natural Log Function?

Definition 1.The function

lnx=? x 11t dt, x >0, is called thenatural logarithm function.•ln1 = 0. •lnx <0 for 0< x <1, lnx >0 forx >1.•d dx (lnx) =1x >0?lnxis increasing.•d 2dx

2(lnx) =-1x

2<0?lnxis concave down.

1.2 Examples

Example 1:lnx= 0and(lnx)?= 1atx= 1Exercise 7.2.23

Show that

lim x→1lnxx-1= 1.Proof. lim x→1lnxx-1= limx→1lnx-ln1x-1=ddx (lnx)????x=1=1x ???x=1= 1.The limit has theindeterminate form?00 ?and is interpreted here in terms of thederivativeof lnx.2

Example 2:lnxandx-1Exercise 7.2.24(a)

Show thatx-1x

lnx=? x 11t dt=1c (x-1).•Ifx >1, then1x <1c <1 andx-1>0 so (1) holds.•If 0< x <1, then 1<1c <1x andx-1<0 so (1) holds.Example 3:lnnand Harmonic NumberExercise 7.2.25(a)

Show that forn≥2

12 +13 +···+1n
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