24 sept 2018 log x. A more refined answer: it looks like a certain integral called ... Proof (Ben): The derivative of logx is 1 x and the derivative of.
LS
1 mar 2016 Strongly convex if ∃α > 0 such that f(x) − α
ORF S Lec gh
Exercise 4 Prove the Laws of Exponents. Hint: make use of the fact that log x is a 1-1 function. Derivatives of the Exponential Function. We already
chapter
New sharp bounds for the logarithmic function
5 mar 2019 In this paper we present new sharp bounds for log(1 + x). We prove that our upper bound is sharper than all the upper bounds presented ...
13 feb 2019 x log x for all x ≥ 17. (1). January 2014]. PROOF OF THE SHELDON CONJECTURE ... Letting z = log x this derivative is z + log z + 1/z.
sheldon
Lemma 1. If x> 1 then. 0<log (r(x))-{(x-$) log (x)-x+i log (2«). Proof. Proof. We prove that for JC^6 the second derivative of h(x) is negative.
div class title some inequalities involving span class italic r span span class sup span class italic r span span div
log p ps . Thus without justification of proving that the derivative of a series Denote the sum over the logarithm of primes by θ (x) = ∑ p≤x log p.
Notes
log. = x b. 6. logb b = 1 and logb 1 = 0. (iv) The derivative of ex Example 9 If ex + ey = ex+y prove that ... Example 13 If xy = ex–y
leep
6 abr 2016 log f(X
Fisher
zk) (from Cauchy-Schwarz inequality) geometric mean: f(x)=(∏ n k=1 xk). 1/n on R n. ++ is concave. (similar proof as for log-sum-exp). Convex functions.
functions
213376
Chapter 8
Logarithms and Exponentials:logx
ande x These two functions are ones with which youalready have some familiarity. Both are in- troduced in many high school curricula, as they have widespread applications in both the scientific and financial worlds. In fact, as recently as 50 years ago, many high school math- ematics curricula included considerable study of "Tables of the Logarithm Function" ("log tables"), because this was prior to the invention of the hand-held calculator. During the Great Depression of the 1930"s, many out-of-work mathematicians and scientists were em- ployed as "calculators" or "computers" to develop these tables by hand, laboriously using difference equations, entry by entry! Here,we are going to use our knowledge of the Fun- damental Theorem of Calculus and the Inverse Function Theorem to develop the properties of the Logarithm Function and Exponential Function. Of course, we don"t need tables of these functions any more because it is possible to buy a hand-held electronic calculator for as little as $10.00, which will compute any value of these functions to 10 decimal places or more! 1
2CHAPTER 8. LOGARITHMS AND EXPONENTIALS:LOGXANDE
X
8.1 The Logarithm Function
Define log(x) (which we shall be thinking of as the natural logarithm) by the following:
Definition 8.1
log(x)= x 1 1 tdtforx>0. Theorem 8.1logxis defined for allx>0.It is everywhere differentiable, hence continuous, and is a 1-1 function. The Range oflogxis(-∞,∞). Proof: Note that forx>0, logxis well-defined, because 1/tis continuous on the interval [1,x](ifx>1) or [x,1] (if 0
0, so logxis increasing (Why?). We postpone the proof of the statement about the Range of logxuntil a bit later. Theorem 8.2 (Laws of Logarithms)(from which we shall subsequently derive the fa- mous "Laws of Exponents"): For all positivex,y, 1.logxy=logx+logy
2.log1/x=-logx
3.logx
r =rlogxfor rationalr. 4.logx
y=logx-logy. Proof: To prove (1), fixyand compute
d dxlogxy=1xyddx(xy)=1xyy=1x=ddxlogx. Then logxyand logxhave the same derivative, from which it follows by the Corollary to the Mean Value Theorem that these two functions differ by a constant: logxy=logx+c. 8.1. THE LOGARITHM FUNCTION3
To evaluatec,letx=1.Since log1 = 0, (why?)c=logy, which proves (1). Toprove(2),weusethesameidea:
d dxlog1x=11/xddx 1 x =1 1/x(-1/x
2 )=-1 x=-ddxlogx, from which it follows (why?) that log 1 x=-logx+c. Again, to evaluatec,letx= 1, and observe thatc= 0, which proves (2). To prove (3),
d dxlogx r =1 x r d dxx r =1 x r rx r-1 =r x=rddxlogx. It follows that
logx r
Chapter 8
Logarithms and Exponentials:logx
ande x These two functions are ones with which youalready have some familiarity. Both are in- troduced in many high school curricula, as they have widespread applications in both the scientific and financial worlds. In fact, as recently as 50 years ago, many high school math- ematics curricula included considerable study of "Tables of the Logarithm Function" ("log tables"), because this was prior to the invention of the hand-held calculator. During the Great Depression of the 1930"s, many out-of-work mathematicians and scientists were em- ployed as "calculators" or "computers" to develop these tables by hand, laboriously using difference equations, entry by entry! Here,we are going to use our knowledge of the Fun- damental Theorem of Calculus and the Inverse Function Theorem to develop the properties of the Logarithm Function and Exponential Function. Of course, we don"t need tables of these functions any more because it is possible to buy a hand-held electronic calculator for as little as $10.00, which will compute any value of these functions to 10 decimal places or more! 1 2CHAPTER 8. LOGARITHMS AND EXPONENTIALS:LOGXANDE
X 8.1 The Logarithm Function
Define log(x) (which we shall be thinking of as the natural logarithm) by the following: Definition 8.1
log(x)= x 1 1 tdtforx>0. Theorem 8.1logxis defined for allx>0.It is everywhere differentiable, hence continuous, and is a 1-1 function. The Range oflogxis(-∞,∞). Proof: Note that forx>0, logxis well-defined, because 1/tis continuous on the interval [1,x](ifx>1) or [x,1] (if 00, so logxis increasing (Why?). We postpone the proof of the statement about the Range of logxuntil a bit later. Theorem 8.2 (Laws of Logarithms)(from which we shall subsequently derive the fa- mous "Laws of Exponents"): For all positivex,y, 1.logxy=logx+logy
2.log1/x=-logx
3.logx
r =rlogxfor rationalr. 4.logx
y=logx-logy. Proof: To prove (1), fixyand compute
d dxlogxy=1xyddx(xy)=1xyy=1x=ddxlogx. Then logxyand logxhave the same derivative, from which it follows by the Corollary to the Mean Value Theorem that these two functions differ by a constant: logxy=logx+c. 8.1. THE LOGARITHM FUNCTION3
To evaluatec,letx=1.Since log1 = 0, (why?)c=logy, which proves (1). Toprove(2),weusethesameidea:
d dxlog1x=11/xddx 1 x =1 1/x(-1/x
2 )=-1 x=-ddxlogx, from which it follows (why?) that log 1 x=-logx+c. Again, to evaluatec,letx= 1, and observe thatc= 0, which proves (2). To prove (3),
d dxlogx r =1 x r d dxx r =1 x r rx r-1 =r x=rddxlogx. It follows that
logx r