Respuesta :
Are you that incapable of pressing some keys on your calculator?
Try it! Take any scientific calculator, turn it on, press the "AC" key, then press "5", "2", "ln" and read off the result of 3.9512437
Try it! Take any scientific calculator, turn it on, press the "AC" key, then press "5", "2", "ln" and read off the result of 3.9512437
3.951243719
The "ln" function which stands for natural logarithm is available in almost every scientific calculator out there and in almost every spreadsheet program. You can also use logarithm tables if you really want to go old school. In any case the value for ln(52) is approximately 3.95124371858143
You can also use the Taylor series to calculate the natural log. The series is
ln(1 + x) = x^1/1 - x^2/2 + x^3/3 - x^4/4 + ...+ ((-1)^(n+1))x^n/n
where x is in the range (-1,1] and n ranges from 1 to infinity.
In this case however, x is outside that range, so you would need to do some division to get the value down to something reasonable. And that's easy. Just keep dividing by e which is:
2.71828182845905
So
52/e = 19.129730940915
19.129730940915/e = 7.03743472830386
7.03743472830386/e = 2.58892755512893
2.58892755512893/e = 0.952413222214178
So after dividing by e 4 times we have a value of 0.952413222214178
and need to solve the equation
1 + x = 0.952413222214178
which is solved with an x value of -0.047586777785822400
Plugging that value into the Taylor series mentioned above and adding 4 (the number of times we divided by e) gives the following progressively more accurate results.
1 term = 3.95241322221418, error = 0.03%
2 terms = 3.95128097150416, error = 0.00094%
3 terms = 3.95124505139554, error = 0.0000337%
4 terms = 3.95124376940386, error = 0.00000129%
5 terms = 3.95124372059918, error = 0.0000000511%
6 terms = 3.9512437186638, error = 0.00000000208%
7 terms = 3.95124371858486, error = 0.0000000000869%
8 terms = 3.95124371858157, error = 0.00000000000368%
9 terms = 3.95124371858143, error = 0.000000000000157%
After this point the terms are too small relative to the sum to have any more effect. So the calculation is completed to the precision of the math used in my spreadsheet.
