2. Assessment (1) Assessment Mathematics: Monday 13 September, 8:30 (last name A-K) or 10:30 (last name L-Z), room 6215 Is everyone registered (cfr. list yesterday?) Please bring: pen ruler scientific calculator (ask me now if you have to borrow one!) (optional) snack and/or drink

3. Assessment (2) After the assessment you sign up for a meeting with the mathematics lecturers on Tuesday. We will discuss your result and give advice about the most appropriate course package for you.

4. Logarithms and exponential equations

5. Logarithms: introduction At the roulette, a person stakes 1 Euro on his favourite number 13. As long as his number does not win, he doubles his stake. At a certain moment we see him stake 1024 Euro. How many times has he played and lost? 1 time lost: stake is 1  2 = 2 1 (Euro) 2 times lost: stake is 2 1  2 = 2 2 (Euro) … etc. x times lost: stake is 2 x (euro) Hence: 2 x = 1024 and so …x = 10 since 2 10 = 1024 New “mathematical operation” needed to find x, PICK THE EXPONENT OF 2 FROM 1024: x = pickexp 2 1024 = 10notation: x = log 2 1024 = 10

6. Logarithms: in general Example of introduction: x = log 2 1024 is the same as x = pickexp 2 1024 and since 1024 = 2 10 we have x = 10 Another example: x = log 5 125 is the same as… x = pickexp 5 125 and since 125 = 5 3 we have x = 3 Exercise 1 in general: y = log g x means x = g y (g > 0, g  1 en x > 0)

7. Special bases example: log 10 000 = log 10 10 000 = log 10 10 4 = 4 g = 10: decimal or common logarithm: log 10 = log g = e = 2.71…: natural logarithm: log e = ln example: ln (1/e 3 )= log e e  3 =  3 Both are on the calculator! Exercises 2 and 3

8. Rules for logarithms (1) Example: In general: (g > 0, g  1 en x 1, x 2 > 0)

9. Rules for logarithms (2) Example: In general: (g > 0, g  1 en x 1, x 2 > 0)

10. Rules for logarithms (3) Example: In general: (g > 0, g  1 and x > 0, r any number)

11. “Rule” (!) for logarithms THAT IS NOT A RULE Example: In general: Hence:

12. Exponential equations (1) A capital of 10 000 Euro is invested at a compound interest rate of 10% per year. How long does it take to double this amount? after 1 year: 10 000  1.10, after 2 years: 10 000  1.10  1.10, … after t years: 10 000  1.10 t Hence we must have: The unknown t is in the exponent: EXPONENTIAL EQUATION.

13. Exponential equations (2) Solving the equation divide LHS and RHS by 10 000 take the logarithm of LHS and RHS rule (3) divide LHS and RHS by log 1.1 The amount will be doubled after 7 years and 3 months.

14. Exercises Exercises 4-8 TO LEARN MATHEMATICS = TO DO A LOT OF EXERCISES YOURSELF, UNDERSTAND MISTAKES AND DO THE EXERCISES AGAIN CORRECTLY