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Math Calculus Calculus & Its Applications (14th Edition)

Simplify the following expression. e 2 ln x

Simplify the following expression. e 2 ln x

Solution Summary: The author explains how to determine the simplification of e2-mathrmln's .

Calculus & Its Applications (14th Edition)

Calculus & Its Applications (14th Edition)

14th Edition

ISBN: 9780134437774

Author: Larry J. Goldstein, David C. Lay, David I. Schneider, Nakhle H. Asmar

Publisher: PEARSON

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Textbook Question

Chapter 4.4, Problem 13E

Simplify the following expression.

e 2 ln x

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Chapter 4.4, Problem 12E

Chapter 4.4, Problem 14E

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A/ Rewrite the following expression in terms of exponentials . Write the final result as simply as you can ? Cosh5 x +Sinh5 x+2sec5x B/Solve the equation for x : Cosh x = Sinh x +1/2.

r1.25 e da /0.25

4

Chapter 4 Solutions

Calculus & Its Applications (14th Edition)

Ch. 4.1 - Can a function such as f(x)=53x be written in the...Ch. 4.1 - Solve the equation 7263x=28.Ch. 4.1 - Prob. 1E Ch. 4.1 - Prob. 2E Ch. 4.1 - Write each expression in Exercises 1-14 in the...Ch. 4.1 - Write each expression in Exercises 1-14 in the...Ch. 4.1 - Write each expression in Exercises 1-14 in the...Ch. 4.1 - Write each expression in Exercises 1-14 in the...Ch. 4.1 - Prob. 7E Ch. 4.1 - Write each expression in Exercises 1-14 in the...

Ch. 4.1 - Write each expression in Exercises 1-14 in the...Ch. 4.1 - Prob. 10E Ch. 4.1 - Prob. 11E Ch. 4.1 - Write each expression in Exercises 1-14 in the...Ch. 4.1 - Prob. 13E Ch. 4.1 - Prob. 14E Ch. 4.1 - Find a number b such that the function f(x)=32x...Ch. 4.1 - Find b so that 8x/3=bx for all x.Ch. 4.1 - Solve the following equations for x. 52x=52 Ch. 4.1 - Solve the following equations for x. 10x=102 Ch. 4.1 - Solve the following equations for x....Ch. 4.1 - Solve the following equations for x....Ch. 4.1 - Solve the following equations for x. 101x=100 Ch. 4.1 - Solve the following equations for x. 24x=8 Ch. 4.1 - Solve the following equations for x. 3(2.7)5x=8.1 Ch. 4.1 - Solve the following equations for x....Ch. 4.1 - Solve the following equations for x. (2x+123)2=2 Ch. 4.1 - Solve the following equations for x. (32x32)4=3 Ch. 4.1 - Solve the following equations for x. 23x=425x Ch. 4.1 - Solve the following equations for x. 35x3x3=0 Ch. 4.1 - Solve the following equations for x. (1+x)2x52x=0 Ch. 4.1 - Prob. 30E Ch. 4.1 - Solve the following equations for x. 2x822x=0 Ch. 4.1 - Prob. 32E Ch. 4.1 - Solve the following equations for x. [Hint: In...Ch. 4.1 - Prob. 34E Ch. 4.1 - Solve the following equations for x. [Hint: In...Ch. 4.1 - Prob. 36E Ch. 4.1 - The expressions in Exercises 37-42 may be factored...Ch. 4.1 - The expressions in Exercises 37-42 may be factored...Ch. 4.1 - The expressions in Exercises 37-42 may be factored...Ch. 4.1 - The expressions in Exercises 37-42 may be factored...Ch. 4.1 - The expressions in Exercises 37-42 may be factored...Ch. 4.1 - Prob. 42E Ch. 4.1 - Prob. 43E Ch. 4.1 - Prob. 44E Ch. 4.1 - Prob. 45E Ch. 4.2 - Solve the following equation for x: e6x=e3.Ch. 4.2 - Differentiate y=(x+ex)4 Ch. 4.2 - Show that ddx(3x)|x=01.1 by calculating the slope...Ch. 4.2 - Show that ddx(2.7x)|x=0.99 by calculating the...Ch. 4.2 - In Exercises 3-6, compute the given derivatives...Ch. 4.2 - Prob. 4E Ch. 4.2 - Prob. 5E Ch. 4.2 - Prob. 6E Ch. 4.2 - Write each expression in the form ekx for a...Ch. 4.2 - Write each expression in the form ekx for a...Ch. 4.2 - Write each expression in the form ekx for a...Ch. 4.2 - Write each expression in the form ekx for a...Ch. 4.2 - Write each expression in the form ekx for a...Ch. 4.2 - Prob. 12E Ch. 4.2 - Solve each equation for x. e5x=e20 Ch. 4.2 - Prob. 14E Ch. 4.2 - Solve each equation for x. ex22x=e8 Ch. 4.2 - Prob. 16E Ch. 4.2 - Solve each equation for x. ex(x21)=0 Ch. 4.2 - Solve each equation for x. 4ex(x2+1)=0 Ch. 4.2 - Find an equation of the tangent line to the graph...Ch. 4.2 - Prob. 20E Ch. 4.2 - Use the first and second derivative rules from...Ch. 4.2 - Prob. 22E Ch. 4.2 - Suppose that A=(a,b) is a point on the graph of...Ch. 4.2 - Find the slope-point form of the equation of the...Ch. 4.2 - Differentiate the following functions. y=3ex7x Ch. 4.2 - Differentiate the following functions. y=2x+45ex4 Ch. 4.2 - Differentiate the following functions. y=xex Ch. 4.2 - Differentiate the following functions....Ch. 4.2 - Differentiate the following functions....Ch. 4.2 - Differentiate the following functions....Ch. 4.2 - Differentiate the following functions. y=exx+1 Ch. 4.2 - Prob. 32E Ch. 4.2 - Differentiate the following functions. y=ex1ex+1 Ch. 4.2 - Differentiate the following functions. y=ex+1 Ch. 4.2 - The graph of y=xex has one extreme point. Find its...Ch. 4.2 - Prob. 36E Ch. 4.2 - Find the point on the graph of y=(1+x2)ex where...Ch. 4.2 - Prob. 38E Ch. 4.2 - Find the slope of the tangent line to the curve...Ch. 4.2 - Find the slope of the tangent line to the curve...Ch. 4.2 - Find the equation of the tangent line to the curve...Ch. 4.2 - Find the equation of the tangent line to the curve...Ch. 4.2 - Find the first and second derivatives....Ch. 4.2 - Find the first and second derivatives. f(x)=exx Ch. 4.2 - Compute the following derivatives. ddx(5ex)...Ch. 4.2 - Prob. 46E Ch. 4.2 - Prob. 47E Ch. 4.2 - Prob. 48E Ch. 4.2 - Prob. 49E Ch. 4.2 - Prob. 50E Ch. 4.2 - Prob. 51E Ch. 4.2 - Prob. 52E Ch. 4.2 - Prob. 53E Ch. 4.2 - Prob. 54E Ch. 4.2 - Prob. 55E Ch. 4.2 - Prob. 56E Ch. 4.3 - Differentiate tet2 Ch. 4.3 - Differentiate [ e3x(1+e6x) ]12.Ch. 4.3 - Differentiate the following functions. f(x)=e2x+3 Ch. 4.3 - Differentiate the following functions. f(x)=e3x2 Ch. 4.3 - Differentiate the following functions. f(x)=e4x2x Ch. 4.3 - Differentiate the following functions....Ch. 4.3 - Differentiate the following functions. f(x)=eex Ch. 4.3 - Differentiate the following functions. f(x)=e1x Ch. 4.3 - Differentiate the following functions. f(x)=ex Ch. 4.3 - Differentiate the following functions. f(x)=ex2+1 Ch. 4.3 - Differentiate the following functions. f(x)=7ex7 Ch. 4.3 - Differentiate the following functions. f(x)=10ex25 Ch. 4.3 - Differentiate the following functions....Ch. 4.3 - Differentiate the following functions....Ch. 4.3 - Differentiate the following functions....Ch. 4.3 - Differentiate the following functions....Ch. 4.3 - Differentiate the following functions....Ch. 4.3 - Differentiate the following functions. f(x)=eeex Ch. 4.3 - Differentiate the following functions....Ch. 4.3 - Differentiate the following functions....Ch. 4.3 - Differentiate the following functions. f(x)=ex+1 Ch. 4.3 - Differentiate the following functions. f(x)=eex Ch. 4.3 - In Exercises 21-26, simplify the function before...Ch. 4.3 - In Exercises 21-26, simplify the function before...Ch. 4.3 - In Exercises 21-26, simplify the function before...Ch. 4.3 - In Exercises 21-26, simplify the function before...Ch. 4.3 - In Exercises 21-26, simplify the function before...Ch. 4.3 - In Exercises 21-26, simplify the function before...Ch. 4.3 - In Exercises 27-32, find the values of x at which...Ch. 4.3 - In Exercises 27-32, find the values of x at which...Ch. 4.3 - In Exercises 27-32, find the values of x at which...Ch. 4.3 - In Exercises 27-32, find the values of x at which...Ch. 4.3 - In Exercises 27-32, find the values of x at which...Ch. 4.3 - In Exercises 27-32, find the values of x at which...Ch. 4.3 - An Investment Portfolio The value of an investment...Ch. 4.3 - Depreciation of Assets The value of the computer t...Ch. 4.3 - The Most Expensive Artwork to Date The highest...Ch. 4.3 - Appreciation of Assets A painting purchased in...Ch. 4.3 - Velocity and Acceleration The velocity of the...Ch. 4.3 - Velocity and Acceleration Suppose the velocity of...Ch. 4.3 - Heights of a Plant The height of a certain plant,...Ch. 4.3 - Heights of a Plant The length of a certain weed,...Ch. 4.3 - Gompertz Growth Curve Let aandb be positive...Ch. 4.3 - Find dydx if y=e(110)ex2.Ch. 4.3 - Size of Tumor In a study, a cancerous tumor was...Ch. 4.3 - Height of a Plant Let f(t) be the function from...Ch. 4.4 - Find lne.Ch. 4.4 - Solve e3x=2 using the natural logarithm function.Ch. 4.4 - Find ln(e).Ch. 4.4 - Find ln(1e2).Ch. 4.4 - If ex=5, Write x in terms of the natural...Ch. 4.4 - If ex=3.2, Write x in terms of the natural...Ch. 4.4 - If lnx=1, Write x using the exponential function.Ch. 4.4 - If lnx=4.5, Write x using the exponential...Ch. 4.4 - Simplify the following expression. lne3 Ch. 4.4 - Simplify the following expression. eln4.1 Ch. 4.4 - Simplify the following expression. eeln1 Ch. 4.4 - Simplify the following expression. ln(e2lne)Ch. 4.4 - Simplify the following expression. ln(lne)Ch. 4.4 - Simplify the following expression. e4ln1 Ch. 4.4 - Simplify the following expression. e2lnx Ch. 4.4 - Simplify the following expression. exln2 Ch. 4.4 - Simplify the following expression. e2ln7 Ch. 4.4 - Simplify the following expression. e2ln7 Ch. 4.4 - Simplify the following expression. elnx+ln2 Ch. 4.4 - Simplify the following expression. eln32lnx Ch. 4.4 - Solve the following equations for x. e2x=5 Ch. 4.4 - Solve the following equations for x. e13x=4 Ch. 4.4 - Solve the following equations for x. ln(4x)=12 Ch. 4.4 - Prob. 22E Ch. 4.4 - Solve the following equations for x. lnx2=9 Ch. 4.4 - Prob. 24E Ch. 4.4 - Solve the following equations for x. 6e0.00012x=3 Ch. 4.4 - Prob. 26E Ch. 4.4 - Solve the following equations for x. ln3x=ln5 Ch. 4.4 - Prob. 28E Ch. 4.4 - Solve the following equations for x. ln(ln3x)=0 Ch. 4.4 - Prob. 30E Ch. 4.4 - Solve the following equations for x. 2ex/39=0 Ch. 4.4 - Prob. 32E Ch. 4.4 - Prob. 33E Ch. 4.4 - Prob. 34E Ch. 4.4 - Prob. 35E Ch. 4.4 - Prob. 36E Ch. 4.4 - Solve the following equations for x. 4exe2x=6 Ch. 4.4 - Prob. 38E Ch. 4.4 - The graph of f(x)=5x+ex is shown in fig. 4. Find...Ch. 4.4 - Prob. 40E Ch. 4.4 - Prob. 41E Ch. 4.4 - Prob. 42E Ch. 4.4 - Prob. 43E Ch. 4.4 - Find the x-intercept of y=(x1)2ln(x+1),x1.Ch. 4.4 - In Exercise 45- 46, find the coordinates of each...Ch. 4.4 - In Exercise 45- 46, find the coordinates of each...Ch. 4.4 - Solve for t. e0.05t4e0.06t=0 Ch. 4.4 - Solve for t. 4e0.01t3e0.04t=0 Ch. 4.4 - Prob. 49E Ch. 4.4 - Wind Velocity Under certain geographic conditions,...Ch. 4.4 - Prob. 51E Ch. 4.4 - Prob. 52E Ch. 4.4 - Prob. 53E Ch. 4.4 - Prob. 54E Ch. 4.4 - Prob. 55E Ch. 4.5 - Differentiate f(x)=1ln(x4+5).Ch. 4.5 - Differentiate f(x)=ln(lnx).Ch. 4.5 - Differentiate the following functions. y=3lnx+ln2 Ch. 4.5 - Differentiate the following functions. y=lnxln3 Ch. 4.5 - Differentiate the following functions. y=x2lnx2 Ch. 4.5 - Differentiate the following functions. y=3lnxx Ch. 4.5 - Differentiate the following functions. y=exlnx Ch. 4.5 - Differentiate the following functions. y=e1+lnx Ch. 4.5 - Differentiate the following functions. y=lnxx Ch. 4.5 - Prob. 8E Ch. 4.5 - Differentiate the following functions. y=lnx2 Ch. 4.5 - Prob. 10E Ch. 4.5 - Differentiate the following functions. y=ln(1x)Ch. 4.5 - Prob. 12E Ch. 4.5 - Differentiate the following functions. y=ln(3x4x2)Ch. 4.5 - Prob. 14E Ch. 4.5 - Differentiate the following functions. y=1lnx Ch. 4.5 - Differentiate the following functions. y=lnxln2x Ch. 4.5 - Differentiate the following functions. y=lnxln2x Ch. 4.5 - Differentiate the following functions. y=(lnx)2 Ch. 4.5 - Differentiate the following functions....Ch. 4.5 - Differentiate the following functions....Ch. 4.5 - Find the second derivatives. d2dt2(t2lnt)Ch. 4.5 - Find the second derivatives. d2dt2ln(lnt)Ch. 4.5 - The graph of f(x)=(lnx)/x is shown in Fig.4. Find...Ch. 4.5 - The graph of f(x)=x/(lnx+x) is shown in Fig.5....Ch. 4.5 - Write the equation of the tangent line to the...Ch. 4.5 - The function f(x)=(lnx+1)/x has a relative extreme...Ch. 4.5 - Determine the domain of definition of the given...Ch. 4.5 - Find the equations of the tangent lines to the...Ch. 4.5 - Find the coordinates of the relative extreme point...Ch. 4.5 - Repeat the previous exercise with y=xlnx.Ch. 4.5 - The graphs of y=x+lnx and y=ln2x are shown in...Ch. 4.5 - Prob. 32E Ch. 4.5 - Prob. 33E Ch. 4.5 - The function y=2x2ln4x (x0) has one minimum point....Ch. 4.5 - A Demand Equation If the demand equation for a...Ch. 4.5 - Total Revenue Suppose that the total revenue...Ch. 4.5 - An Area ProblemFind the maximum area of a...Ch. 4.5 - Analysis of the Effectiveness of an Insect...Ch. 4.6 - Differentiate f(x)=ln[ exx(x+1)6 ].Ch. 4.6 - Use logarithmic differentiation to differentiate...Ch. 4.6 - Simplify the following expressions. ln5+lnx Ch. 4.6 - Simplify the following expressions. lnx5lnx3 Ch. 4.6 - Simplify the following expressions. 12ln9 Ch. 4.6 - Simplify the following expressions. 3ln12+ln16 Ch. 4.6 - Simplify the following expressions. ln4+ln6ln12 Ch. 4.6 - Simplify the following expressions. ln2lnx+ln3 Ch. 4.6 - Simplify the following expressions. e2lnx Ch. 4.6 - Simplify the following expressions. 32ln45ln2 Ch. 4.6 - Simplify the following expressions. 5lnx12lny+3lnz Ch. 4.6 - Simplify the following expressions. elnx2+3lny Ch. 4.6 - Simplify the following expressions. lnxlnx2+lnx4 Ch. 4.6 - Prob. 12E Ch. 4.6 - Simplify the following expressions. Which is...Ch. 4.6 - Simplify the following expressions. Which is...Ch. 4.6 - Evaluate the given expressions. Use ln2=.69 and...Ch. 4.6 - Evaluate the given expressions. Use ln2=.69 and...Ch. 4.6 - Evaluate the given expressions. Use ln2=.69 and...Ch. 4.6 - Prob. 18E Ch. 4.6 - Which of the following is the same as 4ln2x? a....Ch. 4.6 - Prob. 20E Ch. 4.6 - Which of the following is the same as ln8x2ln2x?...Ch. 4.6 - Which of the following is the same as ln9x2? a....Ch. 4.6 - Solve the given equation for x. lnxlnx2+ln3=0 Ch. 4.6 - Solve the given equation for x. lnx2ln3=0 Ch. 4.6 - Solve the given equation for x. lnx42lnx=1 Ch. 4.6 - Solve the given equation for x. lnx2ln2x+1=0 Ch. 4.6 - Solve the given equation for x. (lnx)21=0 Ch. 4.6 - Solve the given equation for x. 3lnxln3x=0 Ch. 4.6 - Solve the given equation for x. lnx=lnx Ch. 4.6 - Solve the given equation for x. 2(lnx)2+lnx1=0 Ch. 4.6 - Solve the given equation for x. ln(x+1)ln(x2)=1 Ch. 4.6 - Solve the given equation for x....Ch. 4.6 - Differentiate. y=ln[(x+5)(2x1)(4x)]Ch. 4.6 - Differentiate. y=ln[(x+1)(2x+1)(3x+1)]Ch. 4.6 - Differentiate. y=ln[(1+x)2(2+x)3(3+x)4]Ch. 4.6 - Differentiate. y=ln[e2x(x3+1)(x4+5x)]Ch. 4.6 - Differentiate. y=ln[xex2+1]Ch. 4.6 - Prob. 38E Ch. 4.6 - Differentiate. y=ln(x+1)4ex1 Ch. 4.6 - Differentiate. y=ln(x+1)4(x3+2)x1 Ch. 4.6 - Prob. 41E Ch. 4.6 - Prob. 42E Ch. 4.6 - Use logarithmic differentiation to differentiate...Ch. 4.6 - Use logarithmic differentiation to differentiate...Ch. 4.6 - Use logarithmic differentiation to differentiate...Ch. 4.6 - Use logarithmic differentiation to differentiate...Ch. 4.6 - Prob. 47E Ch. 4.6 - Use logarithmic differentiation to differentiate...Ch. 4.6 - Use logarithmic differentiation to differentiate...Ch. 4.6 - Use logarithmic differentiation to differentiate...Ch. 4.6 - Prob. 51E Ch. 4.6 - Prob. 52E Ch. 4.6 - Prob. 53E Ch. 4.6 - Prob. 54E Ch. 4 - State as many laws of exponents as you can recall.Ch. 4 - Prob. 2CCE Ch. 4 - Prob. 3CCE Ch. 4 - Prob. 4CCE Ch. 4 - Prob. 5CCE Ch. 4 - Prob. 6CCE Ch. 4 - Prob. 7CCE Ch. 4 - Prob. 8CCE Ch. 4 - Prob. 9CCE Ch. 4 - Prob. 10CCE Ch. 4 - Prob. 11CCE Ch. 4 - Prob. 12CCE Ch. 4 - Prob. 13CCE Ch. 4 - Prob. 14CCE Ch. 4 - Calculate the following. 274/3 Ch. 4 - Calculate the following. 41.5 Ch. 4 - Prob. 3RE Ch. 4 - Prob. 4RE Ch. 4 - Calculate the following. (25/7)14/5 Ch. 4 - Prob. 6RE Ch. 4 - Prob. 7RE Ch. 4 - Calculate the following. 40.240.3 Ch. 4 - Simplify the following. (ex2)3 Ch. 4 - Simplify the following. e5xe2x Ch. 4 - Simplify the following. e3xex Ch. 4 - Simplify the following. 2x3x Ch. 4 - Simplify the following. (e8x+7e2x)e3x Ch. 4 - Simplify the following. e5x/2e3xex Ch. 4 - Solve the following equations for x. e3x=e12 Ch. 4 - Solve the following equations for x. ex2x=e2 Ch. 4 - Solve the following equations for x. (exe2)3=e9 Ch. 4 - Solve the following equations for x. e5xe4=e Ch. 4 - Differntiate the following functions. y=10e7x Ch. 4 - Differntiate the following functions. y=ex Ch. 4 - Differentiate the following functions. y=xex2 Ch. 4 - Differentiate the following functions. y=ex+1x1 Ch. 4 - Differntiate the following functions. y=eex Ch. 4 - Differntiate the following functions. y=(x+1)e2x Ch. 4 - Differentiate the following functions....Ch. 4 - Differentiate the following functions. y=xe Ch. 4 - The graph of the functions f(x)=ex24x2 is shown in...Ch. 4 - Show that the function in Fig. 1 has a relative...Ch. 4 - Solve the following equations for t....Ch. 4 - Solve the following equations for t. et8e0.02t=0 Ch. 4 - Solve the equation 42x=ex. [Hint: Express 2x as an...Ch. 4 - Solve the equation 3x=2ex. [Hint: Express 3x as an...Ch. 4 - Find the points on the graph of y=ex where the...Ch. 4 - Find the points on the graph y=ex+e2x where the...Ch. 4 - Determine the intervals where the function...Ch. 4 - Determine the intervals where the function...Ch. 4 - Find the equation of the tangent line to the graph...Ch. 4 - Show that the tangent lines to the graph of...Ch. 4 - Simplify the following expressions. e(ln5)/2 Ch. 4 - Simplify the following expressions. eln(x2)Ch. 4 - Simplify the following expressions. lnx2lnx3 Ch. 4 - Simplify the following expressions. e2ln2 Ch. 4 - Simplify the following expressions. e5ln1 Ch. 4 - Simplify the following expressions. [elnx]2 Ch. 4 - Solve the following equations for t. tlnt=e Ch. 4 - Solve the following equations for t. ln(ln3t)=0 Ch. 4 - Solve the following equations for t. 3e2t=15 Ch. 4 - Solve the following equations for t. 3et/212=0 Ch. 4 - Solve the following equations for t. 2lnt=5 Ch. 4 - Solve the following equations for t. 2e0.3t=1 Ch. 4 - Differentiate the following functions....Ch. 4 - Differentiate the following functions. y=xlnx Ch. 4 - Differentiate the following functions. y=ln(5x7)Ch. 4 - Differentiate the following functions. y=ln(9x)Ch. 4 - Differentiate the following functions. y=(lnx)2 Ch. 4 - Differentiate the following functions. y=(xlnx)3 Ch. 4 - Differentiate the following functions....Ch. 4 - Differentiate the following functions....Ch. 4 - Differentiate the following functions. y=xlnxx Ch. 4 - Differentiate the following functions. y=e2ln(x+1)Ch. 4 - Differentiate the following functions. y=ln(lnx)Ch. 4 - Differentiate the following functions. y=1lnx Ch. 4 - Differentiate the following functions. y=exlnx Ch. 4 - Differentiate the following functions. y=ln(x2+ex)Ch. 4 - Differentiate the following functions....Ch. 4 - Differentiate the following functions. y=ln|2x+1|Ch. 4 - Differentiate the following functions. y=ln(ex2x)Ch. 4 - Differentiate the following functions. y=lnx3+3x23 Ch. 4 - Differentiate the following functions. y=ln(2x)Ch. 4 - Differentiate the following functions....Ch. 4 - Differentiate the following functions. y=ln|x1|Ch. 4 - Differentiate the following functions....Ch. 4 - Differentiate the following functions. y=ln(1ex)Ch. 4 - Differentiate the following functions....Ch. 4 - Use logarithmic differentiation to differentiate...Ch. 4 - Use logarithmic differentiation to differentiate...Ch. 4 - Use logarithmic differentiation to differentiate...Ch. 4 - Use logarithmic differentiation to differentiate...Ch. 4 - Use logarithmic differentiation to differentiate...Ch. 4 - Prob. 80RE Ch. 4 - Prob. 81RE Ch. 4 - Prob. 82RE Ch. 4 - Use logarithmic differentiation to differentiate...Ch. 4 - Prob. 84RE Ch. 4 - Prob. 85RE Ch. 4 - Prob. 86RE Ch. 4 - Prob. 87RE Ch. 4 - Health Expenditures The health expenditures (in...

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