Chapter 8 Logarithms and Exponentials: logx and e

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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 =rlogx+c, and lettingx=1,weobservec= 0, which proves (3). (Why did we need to requirerto be rational? Why didn"t this prove the theorem for all realr?) 1 (4) Follows from (1) and (2). Theorem 8.3 (Postponed Theorem)Range(logx)=(-∞,∞). Proof: First observe that 1/21dt=1. Now observe that since logxis monotone increasing inx, to compute lim x→0 +logxit suffices 2 to compute the limit along a subsequence ofx"s of our choice, and we choose x n =1/2 n ,n=1,2,... . lim x→0 logx= lim n→∞ log 1 2 n = lim n→∞ -nlog2 =-∞. 1 Because the corollary after Chain Rule only proved differentiation for rational exponents. After we

develop properties of the Exponential Function we will be able to extend (3) to arbitrary real numbersr.

2

See Exercise 1.

4CHAPTER 8. LOGARITHMS AND EXPONENTIALS:LOGXANDE

X

Similarly,

lim x→∞ logx= lim n→∞ log2 n = lim n→∞ nlog2 = +∞.

Exercise 1a. Prove: iffis monotonic, then lim

x→a +f(x) exists if and only if lim n→∞ f(a+ 1/2 n ) exists, and if either of these limits exists, lim x→a

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 =rlogx+c, and lettingx=1,weobservec= 0, which proves (3). (Why did we need to requirerto be rational? Why didn"t this prove the theorem for all realr?) 1 (4) Follows from (1) and (2). Theorem 8.3 (Postponed Theorem)Range(logx)=(-∞,∞). Proof: First observe that 1/21dt=1. Now observe that since logxis monotone increasing inx, to compute lim x→0 +logxit suffices 2 to compute the limit along a subsequence ofx"s of our choice, and we choose x n =1/2 n ,n=1,2,... . lim x→0 logx= lim n→∞ log 1 2 n = lim n→∞ -nlog2 =-∞. 1 Because the corollary after Chain Rule only proved differentiation for rational exponents. After we

develop properties of the Exponential Function we will be able to extend (3) to arbitrary real numbersr.

2

See Exercise 1.

4CHAPTER 8. LOGARITHMS AND EXPONENTIALS:LOGXANDE

X

Similarly,

lim x→∞ logx= lim n→∞ log2 n = lim n→∞ nlog2 = +∞.

Exercise 1a. Prove: iffis monotonic, then lim

x→a +f(x) exists if and only if lim n→∞ f(a+ 1/2 n ) exists, and if either of these limits exists, lim x→a

12345 Next

1. log x derivative formula
2. log x derivative by first principle
3. log x derivative proof
4. log x dérivée
5. log x differentiation
6. log x^2 derivative
7. 1/log x derivative
8. log x^3 derivative

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