stuck to find n with proper math explaination?

Question

0.5^n = n
n= ?

Answer 1

There is no analytical method to solving this. You must use a numerical method.

Let f(n) = n*ln(.5) - ln(n)
We wish to find the value of n for which this function is zero.
f'(n) = ln(.5) - 1/n

Using the newton method, or the bisection method to solve.
Alternatively, you could use MATLAB.
type in the command
fzero('x*log(.5) - log(x)', 0.5)

If you would like to do some analysis to give information on the roots, consider two functions:
a line y = ln(.5) * x = -ln(2) * x
and the curve y = ln(x)
Since the line has negative slope, there must be one point of intersection.
at x = 1, lnx = 0 and -ln(2) * x = -ln2
at x = 0.5, lnx = -0.69, -ln(2) * x = 1.39
So you could see that there must be a root between 0.5 and 1.

Answer 2

Taking ln
n*ln(0.5) = ln(n)
let`stake the function
f(x) = ln x+x*ln2 ( ln(1/2) = --ln2
at x=0 limf(x)=-infinity
limf(x) if x=>+infinity is +infinity
f´(x) = 1/x+ln2 always >0
So the function is increasing and has only one rootwhich is between o and 1 as at x=1 the function is >0

Answer 3

The answer is one.

Why.

The exponent is another way of saying repeated multiplications.

0.5^2 = 0.5 x 0.5.

But 0.5 ^1 means 0.5 (by definition).

Answer 4

0.5^n = n

n = .641186 by trial and error.

Answer 5

i think u use logarithm