SOLUTION: The US gross domestic product GDP in trillions of dollars is approximated by: f(x)= -1.2155=3.7314 Lnx where x=5 corresponds to 1995. When will the GDP reach $12 Trillion? Tha

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Question 93164: The US gross domestic product GDP in trillions of dollars is approximated by:
f(x)= -1.2155=3.7314 Lnx where x=5 corresponds to 1995. When will the GDP reach $12 Trillion?
Thanks in advance for your help
Sorry, this isn't from a text.
Answer by bucky(2189) About Me  (Show Source):
You can put this solution on YOUR website!
Given:
.
f%28x%29+=+-1.2155%2B%283.7314%29%2ALn%28x%29
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[Hope that I interpreted it correctly and the = sign you have between the two terms
on the right side was meant to be a + sign]
.
Set f(x) equal to 12
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12+=+-1.2155+%2B+%283.7314%29%2ALn%28x%29
.
Get rid of the -1.2155 on the right side by adding 1.2155 to both sides:
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13.2155+=+3.7314%2ALn%28x%29
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Solve for Ln(x) by dividing both sides of this equation by 3.7314 to get:
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13.2155%2F3.7314+=+%283.7314%2F3.7314%29%2ALn%28x%29
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which after you do the divisions reduces to:
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3.5417+=+Ln%28x%29
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Just to have things in a little more conventional form, you can transpose the sides of
this equation to get the form:
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Ln%28x%29+=+3.5417
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Now you can solve for x by converting this logarithmic form to the equivalent exponential
form. To do this, you use the rule that says:
.
Log%28b%2CN%29+=+y is equivalent to the exponential form b%5Ey+=+N
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In the natural logarithms b is defined as e (where e is 2.718281828 ...) and from your equation
you find that y = 3.5417 and N = x.
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Now take the exponential form and appropriately substitute the values for b, y, and N to
get that the exponential form of Ln%28x%29=3.5417 is:
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e%5E%283.5417%29+=+x
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But e%5E%283.5417%29 is just a constant that you can get from a scientific calculator.
If you enter 3.5417 and key the e%5Ex function on the calculator you find that the
answer is that x = 34.5256 is the number of years for the GNP to be $12 trillion.
.
Now you need to interpret this. When x is 5, the year is 1995. So x = 34.5256 occurs
29.5245 years after x = 5. Therefore, add 29.5245 years to 1995 and you get 2024.5245.
This means that the economy should reach $12 trillion somewhere near the middle of the year
2024, and it will be slightly above the $12 trillion value at the start of 2025.
.
Check the math here to see if I made any mistakes. The procedures used are correct.
.
Hope this helps you to understand how you can solve the problem.