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Calculus/solving logarithmic function

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Expert: - 9/9/2010


Question
QUESTION: [0.99*ln(1,000,000+x)]+[0.01*ln(900,000+x)]=13.8155. Solve for x.
These are utility functions asking for the minimum price of a insurance premium x. Your help is very much appreciated.

ANSWER: Since the second term would be much smaller than the first,
solve ln(1,000,000+x) = 13.8155.
Do this by making both sides an exponent on e.\

The equation become 1,000,000+x = e^13.8155.
Subtract 1,000,000 from both sides, and there is a guess for x.
Take a number a little higher that x and a number a little lower than x as estimates.  Draw a line between these two points and determine where the line crosses the axis.  This will be the new guess and will replace the guess on the same side of 13.8155 as its value in the function.

This can be repeated as many times as desired.


---------- FOLLOW-UP ----------

QUESTION: Hm..but is there actually a way to solve this? How about replacing the numbers by smaller integers?

Answer
I did it in Excel and approximated it until I got an error of 0.
Now the erroor of 0 is not really 0, but small enough that it might as be.  I'm not sure how accurate Excel was on my PC, but more accurate than what is needed to get a good approximation.

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