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Having finished the work on Casey's computer, you attempt to save the changes and find that this is function is protected by a password. You click the "hint" option and are given the following: What exact value of x satisfies e^2x − 4e^x − 21 = 0 ? If there is no solution, enter NA. Enclose arguments of functions in parentheses and include a multiplication sign between terms. For example, c*log(h) . What value do you enter? x =

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Answer and Step-by-Step Explanation:

To solve the equation e^(2x) - 4e^x - 21 = 0 for x, we can use a substitution method. Let y = e^x.

Substitute y = e^x into the equation:

e^(2x) - 4e^x - 21 = 0

becomes

y^2 - 4y - 21 = 0

Now, we have a quadratic equation in terms of y. We can solve this quadratic equation using the quadratic formula:

y = (-(-4) ± √((-4)^2 - 41(-21))) / (2*1)

y = (4 ± √(16 + 84)) / 2

y = (4 ± √100) / 2

y = (4 ± 10) / 2

So, we have two possible values for y:

y1 = (4 + 10) / 2 = 14 / 2 = 7

y2 = (4 - 10) / 2 = -6 / 2 = -3

Now, remember that y = e^x. So, we can solve for x:

For y = 7:

e^x = 7

x = \ln(7)

For y = -3:

e^x = -3

This has no real solution since e^x is always positive.

Therefore, the value of x that satisfies the equation e^(2x) - 4e^x - 21 = 0 is x = \ln(7).

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