1.Exponents and logarithms (30 points).Aproduction functionis a mathematical modelof the relation between a firm or a country's output and its inputs, which here will be capitaland labor. We'll use the functionY=KL1,(1)whereYis the quantity of output,Kis the quantity of capital (plant and equipment),Listhe quantity of labor (number of employees), and= 1/3 is a parameter (take this as given).We'll setK= 100 andL= 50.a. Compute outputYby entering the following formula in a spreadsheet:= 100^(1/3)*50^(2/3)Make sure you understand what this does. What value do you get? (10 points)b. What is the natural logarithm ofY? (Remember: use the spreadsheet functionLN.)(5 points)c. Show that the production function can be writtenln(Y) =ln(K) + (1) ln(L),where lnxmeans the natural logarithm of the variablex. What properties of logarithms dowe need to derive this from equation (1)? (10 points)d. Compute ln(Y) using the expression you verified in (c), which translates into the spreadsheetformula= (1/3)*LN(100) + (2/3)*LN(50)Verify that your answer is the same as the one you computed in (b)