What real life situations use these functions ( y=ln x ; y= sqrt(x) ; y= x ; y=x^2 ; y=e^x ; y=x^x )?

You can predict when two cars will collide or pass each other if they are moving at a constant speed by using the general equation of a line: y = m(x) + b, at the point of intersection. You can also use this when calculating absorbance of 3 to 5 calibration standards generating a best fit straight line with linear regression. A spreadsheet program can do it easily, and it can be done by your calculator with the statistics functions, as well as calculated by hand (that however, would take me a few hours to explain).

y = x^2. Whenever you throw a ball into the air, it moves in an arc, at least on the earth, and the function of the arc is a parabola: y = a(x)^2 + b(x) + c...pretty sure the a value has to be negative. The quadratic equation occurs in equilibrium calculations.

y = ln(x). Chemists don't like big numbers so we count zeros with logarithms usually base 10 logarithms. The log base e logarithm (or the natural logarithm occurs in nature frequently) and we can inter-convert. Chemists, don't like to say that there are 2.85*10^(-8) protons in solution, we would rather say that the pH = - log[H+] = -log[2.85 * 10^(-8)] = 7.55, or the solution is essentially neutral .

We also use this when calculating the Free Energy Equation: ΔG = RT * ln(k)... where k is the concentration of products over reactants. The more negative delta G is, the more likely the intended reaction will actually occur, and if it does we can speed up this process million-fold by adding biological enzymes to the media.

If you rearrange that equation, you can take out the natural logarithm and express it as e^x functions.

I am not sure about y = x^x, though. I do know that so far it is not possible to integrate y= x^x^x, as even computers cannot do it. I don't know.

I also know that sine and cosine functions occur when measuring the motion of waves on a vibrating string which has applications in music. Also, the whole spectrum of light pulsates between a magnetic field and an electric field at 90 degree angles in a kind of three dimensional sense.