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
In Section 6.1 we introduced the logarithmic functions as inverses of exponential functions We have a power
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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 )
The complex logarithm exponential and power functions
Consider the logarithm of a positive real number. This function satisfies a number of properties: eln x = x. (17) ln(ea) = a
clog
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
Logarithms
The natural logarithm is a logarithm with a specific base: e ≈ 2.71828. This number is called the Euler's number. loge x = ln x. All the properties of the
log
Monday August 31
http://people.hsc.edu/faculty-staff/robbk/Math142/Homework%20Solutions/Assignment%203/HW3.pdf
log logarithmic form: log exponential form: ln logarithmix form: ln
half of text) Section 4.4 Logarithmic Functions. Definition of General Logarithmic Properties of Logarithmic Functions. A. Inverse Properties: ln ln.
MA Lesson Notes
Read Book Natural Logarithm Examples And Answers
9 mars 2022 The logarithmic properties listed above hold for all bases ... Calculus - Derivative of the Natural Log (ln) (worked ... Solving Logarithmic ...
Physics 116AWinter 2011
The complex logarithm, exponential and power functions In these notes, we examine the logarithm, exponential and power functions, where the arguments ?of these functions can be complex numbers. In particular, we are interested in how their properties differ from the properties of thecorresponding real-valued functions.1. Review of the properties of the argument of a complex number
Before we begin, I shall review the properties of the argument of anon-zero complex numberz, denoted by argz(which is a multi-valued function), and the principal valueof the argument, Argz, which is single-valued and conventionally defined such that: Details can be found in the class handout entitled,The argument of a complex number. Here, we recall a number of results from that handout. One can regard argzas a set consisting of the following elements, One can also express Argzin terms of argzas follows:Argz= argz+ 2π?1
2-argz2π?
,(3) where [ ] denotes the greatest integer function. That is, [x] is defined to be the largest integer less than or equal to the real numberx. Consequently, [x] is the unique integer that satisfies the inequality ?Note that the wordargumenthas two distinct meanings. In this context, given a function w=f(z), we say thatzis the argument of the functionf. This should not be confused with the argument of a complex number, argz. †The following three books were particularly useful in the preparation of these notes:1.Complex Variables and Applications, by James Ward Brown and Ruel V. Churchill (McGraw
Hill, New York, 2004).
2.Elements of Complex Variables, by Louis L. Pennisi, with the collaboration of Louis I. Gordon
and Sim Lasher (Holt, Rinehart and Winston, New York, 1963).3.The Theory of Analytic Functions: A Brief Course, by A.I. Markushevich (Mir Publishers,
Moscow, 1983).
1 For example, [1.5] = [1] = 1 and [-0.5] =-1. One can check that Argzas defined in eq. (3) does fall inside the principal interval specified by eq. (1). The multi-valued function argzsatisfies the following properties, arg(z1z2) = argz1+ argz2,(5) arg ?z1 z2? = argz1-argz2.(6) arg ?1 z? = argz=-argz .(7) Eqs. (5)-(7) should be viewed as set equalities, i.e. the elements of the sets indi- cated by the left-hand side and right-hand side of the above identities coincide. However, the following results arenotset equalities: argz+ argz?= 2argz ,argz-argz?= 0,(8) which, by virtue of eqs. (5) and (6), yield: argz2= argz+ argz?= 2argz ,arg(1) = argz-argz?= 0.(9) For example, arg(1) = 2πn, forn= 0±1,±2,.... More generally, argzn= argz+ argz+···argz? n?=nargz .(10) We also note some properties of the the principal value of the argument.Arg (z1z2) = Argz1+ Argz2+ 2πN+,(11)
Arg (z1/z2) = Argz1-Argz2+ 2πN-,(12)
where the integersN±are determined as follows: N±=?????-1,if Argz1±Argz2> π,
If we setz1= 1 in eq. (12), we find that
Arg(1/z) = Arg
z=?Argz ,if Imz= 0 andz?= 0,
-Argz ,if Imz?= 0.(14)Note that forzreal, both 1/zand
zare also real so that in this casez=zandArg(1/z) = Arg
z= Argz. In addition, in contrast to eq. (10), we haveArg(zn) =nArgz+ 2πNn,(15)
2 where the integerNnis given by: N n=?12-n2πArgz?
,(16) and [ ] is the greatest integer bracket function introduced in eq. (4).2. Properties of the real-valued logarithm, exponential and power func-
tions Consider the logarithm of a positive real number. This function satisfies a number of properties: e lnx=x,(17) ln(ea) =a,(18) ln(xy) = ln(x) + ln(y),(19) ln ?x y? = ln(x)-ln(y),(20) ln ?1 x? =-ln(x),(21) lnxp=plnx,(22) for positive real numbersxandyand arbitrary real numbersaandp. Likewise, the power function defined over the real numbers satisfies: x a=ealnx,(23) x axb=xa+b,(24) x a xb=xa-b,(25) 1 xa=x-a,(26) (xa)b=xab,(27) (xy)a=xaya,(28) ?x y? a =xay-a,(29) for positive real numbersxandyand arbitrary real numbersaandb. Closely related to the power function is the generalized exponential function defined over 3 the real numbers. This function satisfies: a x=exlna,(30)Physics 116AWinter 2011
The complex logarithm, exponential and power functions In these notes, we examine the logarithm, exponential and power functions, where the arguments ?of these functions can be complex numbers. In particular, we are interested in how their properties differ from the properties of thecorresponding real-valued functions.1. Review of the properties of the argument of a complex number
Before we begin, I shall review the properties of the argument of anon-zero complex numberz, denoted by argz(which is a multi-valued function), and the principal valueof the argument, Argz, which is single-valued and conventionally defined such that: Details can be found in the class handout entitled,The argument of a complex number. Here, we recall a number of results from that handout. One can regard argzas a set consisting of the following elements, One can also express Argzin terms of argzas follows:Argz= argz+ 2π?1
2-argz2π?
,(3) where [ ] denotes the greatest integer function. That is, [x] is defined to be the largest integer less than or equal to the real numberx. Consequently, [x] is the unique integer that satisfies the inequality ?Note that the wordargumenthas two distinct meanings. In this context, given a function w=f(z), we say thatzis the argument of the functionf. This should not be confused with the argument of a complex number, argz. †The following three books were particularly useful in the preparation of these notes:1.Complex Variables and Applications, by James Ward Brown and Ruel V. Churchill (McGraw
Hill, New York, 2004).
2.Elements of Complex Variables, by Louis L. Pennisi, with the collaboration of Louis I. Gordon
and Sim Lasher (Holt, Rinehart and Winston, New York, 1963).3.The Theory of Analytic Functions: A Brief Course, by A.I. Markushevich (Mir Publishers,
Moscow, 1983).
1 For example, [1.5] = [1] = 1 and [-0.5] =-1. One can check that Argzas defined in eq. (3) does fall inside the principal interval specified by eq. (1). The multi-valued function argzsatisfies the following properties, arg(z1z2) = argz1+ argz2,(5) arg ?z1 z2? = argz1-argz2.(6) arg ?1 z? = argz=-argz .(7) Eqs. (5)-(7) should be viewed as set equalities, i.e. the elements of the sets indi- cated by the left-hand side and right-hand side of the above identities coincide. However, the following results arenotset equalities: argz+ argz?= 2argz ,argz-argz?= 0,(8) which, by virtue of eqs. (5) and (6), yield: argz2= argz+ argz?= 2argz ,arg(1) = argz-argz?= 0.(9) For example, arg(1) = 2πn, forn= 0±1,±2,.... More generally, argzn= argz+ argz+···argz? n?=nargz .(10) We also note some properties of the the principal value of the argument.Arg (z1z2) = Argz1+ Argz2+ 2πN+,(11)
Arg (z1/z2) = Argz1-Argz2+ 2πN-,(12)
where the integersN±are determined as follows: N±=?????-1,if Argz1±Argz2> π,
If we setz1= 1 in eq. (12), we find that
Arg(1/z) = Arg
z=?Argz ,if Imz= 0 andz?= 0,
-Argz ,if Imz?= 0.(14)Note that forzreal, both 1/zand
zare also real so that in this casez=zandArg(1/z) = Arg
z= Argz. In addition, in contrast to eq. (10), we haveArg(zn) =nArgz+ 2πNn,(15)
2 where the integerNnis given by: N n=?12-n2πArgz?
,(16) and [ ] is the greatest integer bracket function introduced in eq. (4).2. Properties of the real-valued logarithm, exponential and power func-
tions Consider the logarithm of a positive real number. This function satisfies a number of properties: e lnx=x,(17) ln(ea) =a,(18) ln(xy) = ln(x) + ln(y),(19) ln ?x y? = ln(x)-ln(y),(20) ln ?1 x? =-ln(x),(21) lnxp=plnx,(22) for positive real numbersxandyand arbitrary real numbersaandp. Likewise, the power function defined over the real numbers satisfies: x a=ealnx,(23) x axb=xa+b,(24) x a xb=xa-b,(25) 1 xa=x-a,(26) (xa)b=xab,(27) (xy)a=xaya,(28) ?x y? a =xay-a,(29) for positive real numbersxandyand arbitrary real numbersaandb. Closely related to the power function is the generalized exponential function defined over 3 the real numbers. This function satisfies: a x=exlna,(30)- log properties ln
- logarithm ln properties