student85 said:
hmmm i don't get you.
But please someone tell me yes or no...is there a way to get an exact value for x manually??
And arildno, what are those techniques, can you please share them with me?
No one can answer your first question until you tell what you mean by "manually"! (Counting on your fingers?
) I would interpret "manually" as meaning "without a calculator" but then how are you going to find the logarithm itself?
A very simple numerical method (a little more sophisticated than "trial and error") is "bisection". First find an interval having a positive value for the function at one end and negative at the other (so you know that, for any continuous function, there is a zero in the interval). Here it is easy to see (using a calculator), with f(x)= xln(x)+ x
2- 10, that f(2) is negative but f(3) is positive so there exist a solution between 2 and 3. Try half way between: calculate f(2.5) and see whether it is positive or negative. If it is positive, then there must be a solution between 2 and 2.5. If it is negative, then there must be a solution between 2.5 and 3.0. In either case, try half way between again. Keep doing that until you have enough accuracy.
"Newton's method" is more sophisticated and more accurate but more difficult to explain- you use the derivative to replace the function, at some starting x, with its tangent line- solve that linear equation for a new "trial" x, and repeat.
By the way, "trial and error", while not very sophisticated, is a well respected mathematical technique!
