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1 Suppose that there s a real number x that satisfies the equation ex = 0 We know that our mystery number x can t be equal to 0, because we can plug x = 0 straightaway into the equation and get e0 = 1 based on the fact that any positive real number (including e) raised to the zeroth power is equal to 1 We ve assumed that x is real, and we ve discovered that x 0, so we can be
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certain that 1/x is a nonzero real number Therefore, it s okay for us to take the (1/x)th power of both sides of the original equation to obtain (ex)(1/x) = 0(1/x) which we can rewrite using the algebraic rules for exponents as e[x(1/x)] = 0(1/x) When 0 is raised to any nonzero real power, the result is 0 That fact, along with another dose of the algebraic rules for exponents, allows us to streamline the above equation, getting e(x /x) = 0 which simplifies further to e1 = 0 and finally to e=0 This statement is patently untrue According to reductio ad absurdum, it follows that our original assumption must be false We must conclude that no real-number power of e is equal to 0 2 The dashed gray curves in Fig B-14 are the graphs of y = ex
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Illustration for the solution to Problem 2 in Chap 14
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540 Worked-Out Solutions to Exercises: 11-19
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and y = e x taken directly from Figs 14-1A and 14-2A Now let s consider the function y = ex e x Using the algebraic rules for exponents, we can rewrite this as y = e[x+( x)] which simplifies to y = e0 and further to y=1 The domain of this constant function encompasses all real numbers The range is the set containing the single element 1 The graph is a solid black horizontal line passing through the point (0,1) in Fig B-14 3 The dashed gray curves in Fig B-15 are the graphs of y = 10x and y = 10 x
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Figure B-15
Illustration for the solution to Problem 3 in Chap 14
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taken directly from Figs 14-1B and 14-2B Now let s look at the ratio function y = 10x /10 x Using the algebraic rules for exponents, we can rewrite this as y = 10[x ( x)] which simplifies to y = 102x The domain encompasses all real numbers The range is the set of all positive real numbers The graph is the solid black curve in Fig B-15 4 Figure B-16 shows the same graphs as Fig B-15 However, in this illustration, the y axis is logarithmic, spanning the three orders of magnitude from 01 to 100 The dashed gray lines are the graphs of y = 10x and y = 10 x
Figure B-16
Illustration for the solution to Problem 4 in Chap 14
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542 Worked-Out Solutions to Exercises: 11-19
The solid black line is the graph of y = 102x 5 Figure B-17 is the graph of the function y = 10(1/x) for values of x ranging from 10 to 10 When we input x = 0, we get 101/0, which is undefined For any other real value of x, the output value y is a positive real, so the domain is the set of all nonzero reals No matter how large we want y to be when y > 1, we can always find some value of x that will produce it No matter how small we want y to be when 0 < y < 1, we can always find some value of x that will produce it However, we can t find any value for x that will give us y = 1 For that to happen, we must raise 10 to the zeroth power, meaning that we must find some x such that 1/x = 0 No such x exists, so the range of the function is the set of all positive reals except 1 The graph has a horizontal asymptote whose equation is y = 1, and a vertical asymptote corresponding to the y axis The open circle at (0,0) tells us that this point is not part of the graph 6 Suppose that there s a real number x that satisfies the equation ln 0 = x
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