logarithm

that many mathematicians uselogxto denote the natural logarithm rather than the logarithm with base10.

properties we saw earlier for logarithms with any base. For review, we summarize

the key properties here. In the box below, we assume

x

and

y

are positive

numbers.

Properties of the natural logarithm

ln 1=0

lne=1

ln(xy)=lnx+lny

ln

_x1

= −lnx

ln

x_y

=lnx−lny

ln(x

t

_)=tlnx

The exponential function

e

x

_{and the natural logarithm lnx}

_{(which equals}

log

_e

x) are the inverse functions for each other, just as the functions 2

x

_and

log

₂

xare the inverse functions for each other. Thus the exponential function

and the natural logarithm exhibit the same behavior as any two functions that

are the inverse functions for each other. For review, we summarize here the key

properties connecting the exponential function and the natural logarithm:

-2 2 4 6 x -2 2 4 6 y

The graph of lnx(red)is obtained by flipping the graph of ex_(blue)across

the liney=x.

Connections between the exponential function and the natural

logarithm

•

lny=xmeanse

x

_=y.

•

ln(e

x

_{)=xfor every real numberx.}

•

e

lny

=yfor every positive numbery.

exercises

The next two exercises emphasize that ln(x+y)does not equallnx+lny.

1 Forx=7 andy=13, evaluate each of the follow- ing:

(a) ln(x+y) (b) lnx+lny

2 Forx = 0.4 andy = 3.5, evaluate each of the following:

(a) ln(x+y) (b) lnx+lny

The next two exercises emphasize thatln(xy)does not equal(lnx)(lny).

3 Forx=3 andy=8, evaluate each of the follow- ing:

(a) ln(xy) (b) (lnx)(lny)

4 Forx=1.1 andy=5, evaluate each of the follow- ing:

(a) ln(xy) (b) (lnx)(lny)

The next two exercises emphasize that ln_yx does not equal ln_lnx_y.

5 Forx=12 andy=2, evaluate each of the follow- ing:

(a) lnx_y (b) ln_lnyx

6 For x = 18 andy = 0.3, evaluate each of the following:

(a) lnx_y (b) ln_lnyx

7 Find a numberysuch that lny=4.

8 Find a numbercsuch that lnc=5.

9 Find a numberxsuch that lnx= −2.

10 Find a numberxsuch that lnx= −3.

11 Find a numbertsuch that ln(2t+1)= −4.

12 Find a numberwsuch that ln(3w−2)=5.

13 Find all numbersysuch that ln(y2_+1)=3.

14 Find all numbersrsuch that ln(2r2_{−3)= −1.}

15 Find a numberxsuch thate3x−1_=2.

16 Find a numberysuch thate4y−3_=5.

For Exercises 17–28, find all numbersxthat satisfy the given equation. 17 ln(x+5)−ln(x−1)=2 18 ln(x+4)−ln(x−2)=3 19 ln(x+5)+ln(x−1)=2 20 ln(x+4)+ln(x+2)=2 21 ln(12x) ln(5x) =2 22 ln(11x) ln(4x) =2 23 e2x_+ex₌₆ 24 e2x_−4ex₌₁₂ 25 ex_+e−x₌₆ 26 ex_+e−x₌₈ 27 ln(3x)_lnx =4 28 ln(6x)_lnx =5

29 Find the numbercsuch that area(1

x,1, c)=2. 30 Find the numbercsuch that area(1

x,1, c)=3. 31 Find the numbertthat makeset2_+6t

as small as pos- sible.

[Hereet2_+6t

meanse(t2_+6t) .]

32 Find the numbertthat makeset2_+8t+3

as small as possible.

33 Find a numberysuch that 1+lny 2+lny =0.9.

34 Find a numberwsuch that 4−lnw

3−5 lnw =3.6.

For Exercises 35–38, find a formula for (f◦g)(x)as- suming thatfandgare the indicated functions. 35 f (x)=lnx and g(x)=e5x

36 f (x)=lnx and g(x)=e4−7x 37 f (x)=e2x and g(x)=lnx

38 f (x)=e8−5x _{and g(x)=lnx}

For each of the functionsf given in Exercises 39–48: (a) Find the domain of f.

(b) Find the range of f. (c) Find a formula forf−1_.

(d) Find the domain of f−1_.

(e) Find the range of f−1_.

You can check your solutions to part (c) by verifying that f−1_{◦f = I and f◦f}−1 _{=I. (Recall that I is the}

function defined byI(x)=x.) 39 f (x)=2+lnx 40 f (x)=3−lnx 41 f (x)=4−5 lnx 42 f (x)= −6+7 lnx 43 f (x)=3e2x 44 f (x)=5e9x 45 f (x)=4+ln(x−2) 46 f (x)=3+ln(x+5) 47 f (x)=5+6e7x 48 f (x)=4−2e8x

49 What is the area of the region under the curvey= 1_x, above thex-axis, and between the linesx =1 and x=e2?

50 What is the area of the region under the curvey= 1_x, above thex-axis, and between the linesx =1 and x=e5?

problems

51 Verify that the last five rectangles in the figure in Example 2 have area 1₈, 1₉, ₁₀1, ₁₁1, and₁₂1.

52 Consider this figure:

y = 1x

1 1.5 2 2.5 x

1 y

The region under the curvey=_x1, above thex-axis, and between the linesx=1andx=2.5.

(a) Calculate the sum of the areas of all six rectan- gles shown in the figure above.

(b) Explain why the calculation you did in part (a) shows that

area(1

x,1,2.5) <1.

(c) Explain why the inequality above shows that e >2.5.

The following notation is used in Problems 53–56: area(x2_{,1,2)is the area of the region under the curve} y =x2, above thex-axis, and between the linesx =1 andx=2, as shown below.

y = x2 -2 -1 1 2 x 1 2 3 4 y

53 Using one rectangle, show that 1<area(x2_,1,2).

54 Using one rectangle, show that area(x2_{,1,2) <4.}

55 Using four rectangles, show that 1.96<area(x2_,1,2).

56 Using four rectangles, show that area(x2_{,1,2) <2.72.}

[The two problems above show that area(x2_,1,2)

is in the interval [1.96,2.72]. If we use the midpoint of that interval as an estimate, we get

area(x2_,1,2)≈ 1.96+2.72

2 =2.34. This is a very good

estimate—the exact value of area(x2_,1,2)is 7 3, which

is approximately2.33.]

57 Explain why

lnx≈2.302585 logx for every positive numberx.

58 Explain why the solution to part (b) of Exercise 5 in this section is the same as the solution to part (b) of Exercise 5 in Section 3.3.

59 Supposecis a number such that area(1

x,1, c) >1000.

Explain whyc >21000_.

The functionscoshandsinhare defined by coshx= e

x_+e−x

2 and sinhx=

ex_−e−x 2 for every real numberx. These functions are called the hyperbolic cosineandhyperbolic sine; they are useful in engineering.

60 Show that cosh is an even function.

61 Show that sinh is an odd function.

62 Show that

(coshx)2_−(sinhx)2₌₁ for every real numberx.

63 Show that coshx≥1 for every real numberx.

64 Show that

cosh(x+y)=coshxcoshy+sinhxsinhy for all real numbersxandy.

65 Show that

sinh(x+y)=sinhxcoshy+coshxsinhy for all real numbersxandy.

66 Show that

(coshx+sinhx)t_{=cosh(tx)+sinh(tx)}

for all real numbersxandt.

67 Show that ifxis very large, then coshx≈sinhx≈ e

x

68 Show that the range of sinh is the set of real numbers.

69 Show that sinh is a one-to-one function and that its inverse is given by the formula

(sinh)−1_{(y)=ln y+}q y2₊₁

for every real numbery.

70 Show that the range of cosh is the interval[1,∞).

71 Supposefis the function defined by f (x)=coshx

for everyx≥0. In other words,f is defined by the same formula as cosh, but the domain off is the interval[0,∞)and the domain of cosh is the set of real numbers. Show thatf is a one-to-one function and that its inverse is given by the formula

f−1_{(y)=ln y+}q y2₋₁

for everyy≥1.

72 Write a description of how the shape of the St. Louis Gateway Arch is related to the graph of coshx. You should be able to find the necessary information us- ing an appropriate web search.

The St. Louis Gateway Arch, the tallest national monument in the United States.

worked-out solutions

to Odd-Numbered Exercises

In document Exponential Functions, Logarithms, and e (Page 61-64)