How do I solve for x ?
#log_ax=x^2/a^2#
1 Answer
Apart from
Explanation:
Given:
#log_a x = x^2/a^2#
Note that one straightforward solution is
#log_a a = 1 = a^2/a^2#
Beyond that simple solution, we need to use the Lambert W function...
We want to get this into a form:
#z e^z = w#
so we can use the Lambert W function.
Using the change of base formula, we can write:
#x^2/a^2 = ln x / ln a = ln x^2 / (2 ln a)#
So:
#(2 ln a)/a^2 x^2 = ln x^2#
Taking the exponent of both sides:
#e^((2 ln a)/a^2 x^2) = x^2#
Multiplying both sides by
#1 = x^2 e^(-(2 ln a)/a^2 x^2)#
Multiplying both sides by
#-(2 ln a)/a^2 = (-(2 ln a)/a^2 x^2) e^(-(2 ln a)/a^2 x^2)#
Applying the Lambert W function to both sides:
#W(-(2 ln a)/a^2) = -(2 ln a)/a^2 x^2#
Multiplying both sides by
#x^2 = -a^2/(2 ln a) W(-(2 ln a)/a^2)#
Then taking the positive square root of both sides, we get:
#x = sqrt(-a^2/(2 ln a) W(-(2 ln a)/a^2))#