The Intermediate value theorem states that if a continuous function, y=f(x) crosses the x-axis between two values of x, then f(x) has a zero (or root) between the two values of x.
For the given problem, define the function
f(x) = cos(x) + ln(x) - x2 + 4
This function should be zero at a certain value of x. Let us use the Intermediate theorem to verify that it is so.
Let x=1. Then f(1) = cos(1) + ln(1) - 12 + 4 = 3.5403
Let x=2. Then f(2) = cos(2) + ln(2) - 22 + 4 = -3.723
Because the value of the f(x) changes sign between x=1 and x=2, there is a zero between x=1 and x=2.
The graphical calculator verifies that the zero occurs at approximately x = 1.642.