Find x > 3 such that
ln(x) < x^(0.1)
To find the values of x that satisfy the inequality ln(x) < x^(0.1), we can start by expressing the inequality as an equation:
ln(x) = x^(0.1)
Now, we can raise both sides of the equation to the power of e (the base of the natural logarithm) to remove the natural logarithm:
e^(ln(x)) = e^(x^(0.1))
Since e^(ln(x)) is equal to x (by the definition of the natural logarithm), the equation simplifies to:
x = e^(x^(0.1))
To solve this equation, we can use a numerical approximation method such as trial and error or graphing. By examining the graph of the function f(x) = e^(x^(0.1)) or plugging in values for x, we can find that x > 3 satisfies the equation.
So, if we restrict our search to values of x greater than 3, we can conclude that x > 3 is the solution to the given inequality ln(x) < x^(0.1).