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Question
Dear Sir,
First of all i would like to thank u for the answer u gave me related to log. Here is another question.
How to solve question using log
2^x + 9^x = 92
If we take log to both sides then
log(2^x + 9^x) = log 92
Here the main problem rises how to open left side of eq.
Is there another method to solve it (i mean not using log)
Pleas guide.
Thnak in advance.
I believe the easiest way to find a solution is to use a spreadsheet and solve for x using
the Bisection Method. Start with the points 1 and 3. This is becasue 2^1 + 9^1 = 2 + 9 = 11,
which is less than 92, and 2^3 + 9^3 = 8 + 729 = 737, which is too high.
It takes a little more work, but the Secant Method could also be used.
It says to choose any two points x1,x2 to start with and calculate x3 by
x3 = x2 - (f(x3)-f(x2))/(x3-x2).
To do it even quicker (though I believe the secant method would work in a half dozen steps or so)
Newton's Method says that x2 = x1 - f(x1)/f'(x1). Only 1 point needs to be used, but f' needs to be found.
Now if f(x) = 92 - 2^x - 9^x, f'(x) = ln(2)2^x + ln(9)2^x.
So the formula to use is x2 = x1 - (92 - 2^(x1) - 9^(x1))/(ln(2)2^(x1) + ln(9)2^(x1)).
I think the Secant Method would be the best. The bisection method could be used,
but takes oveer two dozen steps until an answer is found.
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