Answer:
x = 0.3466
Explanation:
o solve the equation x^x = 1/4, we can take the natural logarithm of both sides and use the laws of logarithms to simplify the expression:
ln(x^x) = ln(1/4)
Using the rule that ln(a^b) = b * ln(a), we get:
x * ln(x) = ln(1/4)
Using the property that ln(1/a) = -ln(a), we can simplify the right-hand side:
x * ln(x) = -ln(4)
Using the fact that ln(e) = 1, we can write ln(4) as 2 * ln(2):
x * ln(x) = -2 * ln(2)
Next, we can divide both sides by ln(x) to isolate x:
x = (-2 * ln(2)) / ln(x)
At this point, we can use numerical methods or a calculator to approximate the value of x. One possible method is to use a graphing calculator to plot the function f(x) = x^x - 1/4 and find its root(s). Another method is using an iterative process, such as Newton's, to find a solution.
Using a graphing calculator, we can see that the equation has one solution near x = 0.3465735903. Therefore, the solution to the equation x^x = 1/4 is approximately x = 0.3466 (rounded to four decimal places).