Properties of Exponents and Logarithms
Properties of Exponents and Logarithms Then the following properties of ... Most calculators can directly compute logs base 10 and the natural log.
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 . & .
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
lecture handout
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
Elementary Functions Rules for logarithms Exponential Functions
We review the properties of logarithms from the previous lecture. In that By the first inverse property since ln() stands for the logarithm base.
. Working With Logarithms (slides to )
LOGARITHME NEPERIEN
Preuve : Les démonstrations se font principalement en utilisant les propriétés de la fonction exponentielle. • e ln a + ln
ln
Algebraic Properties of ln(x)
simplify the natural logarithm of products and quotients. If a and b are positive numbers and r is a rational number we have the following properties:.
Calculating With Logarithms
11.4 Properties of Logarithms
a. The first thing we must do is move the coefficients from the front into the exponents by using property 3. This gives us. 4 ln 2 + 2 ln x – ln y = ln 24
Elementary Functions The logarithm as an inverse function
then the properties of logarithms will naturally follow from our understanding of exponents. ln(x) and speak of the “natural logarithm”.
. Logarithms (slides to )
PROPERTIES OF LOGARITHMIC FUNCTIONS
log is often written as x ln and is called the NATURAL logarithm (note: 59. 7182818284 .2. ≈ e. ). PROPERTIES OF LOGARITHMS. EXAMPLES.
properties of logarithms
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 paper2×2×2×2×2×2,
but what does the computer actually do when we type2^6•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.1What 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 2dx2(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.2Example 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 +···+1nExample 4: Euler"s Constantγ4
Proof.
•The sum of the shaded areas is given by S n=Uf(P)-? n 11t dt= 1 +12 +13 +···+1n-1-lnn.Example 4: Euler"s ConstantγProof. (cont.)
•The sum of the areas of the triangles formed by connecting the points (1,1),···, (n,1n ) is T n=12·1??
1-12 +···+?1n-1-1n =12 1-1n Lecture 2Section 7.2 The Logarithm Function, Part IJiwen 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 paper2×2×2×2×2×2,
but what does the computer actually do when we type2^6•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.1What 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 2dx2(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.2Example 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 +···+1nExample 4: Euler"s Constantγ4
Proof.
•The sum of the shaded areas is given by S n=Uf(P)-? n 11t dt= 1 +12 +13 +···+1n-1-lnn.Example 4: Euler"s ConstantγProof. (cont.)
•The sum of the areas of the triangles formed by connecting the points (1,1),···, (n,1n ) is T n=12·1??
1-12 +···+?1n-1-1n =12 1-1n- log ln properties
- ln properties logarithm
- logarithmic properties ln