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

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.

Logarithms and exponential equations

Logarithms: introductionAt 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 = 21 (Euro)2 times lost: stake is 21 2 = 22 (Euro) … etc.

x times lost: stake is 2x (euro)Hence: 2x = 1024 and so … x = 10 since 210 = 1024New “mathematical operation” needed to find x, PICK THE EXPONENT OF 2 FROM 1024:x = pickexp2 1024 = 10 notation: x = log2 1024 = 10

Logarithms: in general

Example of introduction: x = log2 1024 is the same as x = pickexp2 1024 and since 1024 = 210 we have x = 10

Another example: x = log5 125 is the same as…

x = pickexp5 125 and since 125 = 53 we have x = 3

Exercise 1

in general: y = logg x means x = gy

(g > 0, g 1 en x > 0)

Special bases

example: log 10 000 = log10 10 000 = log10 104 = 4

• g = 10: decimal or common logarithm: log10 = log

• g = e = 2.71…: natural logarithm: loge = ln

example: ln (1/e3)= loge e 3 = 3

Both are on the calculator!

Exercises 2 and 3

Rules for logarithms (1)

Example:

In general: 1 2 1 2log log logg g gx x x x

(g > 0, g 1 en x1, x2 > 0)

328 2 and hence log 8 3

224 2 and hence log 4 2

2 3 2 34 8 2 2 2 2 2 2and hence log 4 8 2 3 log 4 log 8

Rules for logarithms (2)

Example:

In general:

(g > 0, g 1 en x1, x2 > 0)

328 2 and hence log 8 3

224 2 and hence log 4 2

22 3

3

4 22

8 2

2 2 2

4and hence log 2 3 log 4 log 8

8

11 2

2

log log logg g g

xx x

x

Rules for logarithms (3)

Example:

In general: log logrg gx r x

(g > 0, g 1 and x > 0, r any number)

328 2 and hence log 8 3

1 11 335 558 2 2

15

2 2

1 1and hence log 8 3 log 8

5 5

“Rule” (!) for logarithms THAT IS NOT A RULE

Example:

In general:

1 2There is NO SIMPLE RULE for log g x x

52 2 2log 16 16 log 32 log 2 5

4 42 2 2 2log 16 log 16 log 2 log 2 4 4 8

Hence: 2 2 2log 16 16 log 16 log 16

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 1.10t

Hence we must have: 10 000 1.10 20 000t

The unknown t is in the exponent: EXPONENTIAL EQUATION.

Exponential equations (2)

Solving the equation 10 000 1.1 20 000 :t

10 000 1.1 20 000t divide LHS and RHS by 10 000

1.1 2t take the logarithm of LHS and RHS

log1.1 log 2t rule (3)

log1.1 log 2t divide LHS and RHS by log 1.1

log 27.27...

log1.1t

The amount will be doubled after 7 years and 3 months.

Exercises

• Exercises 4-8

TO LEARN MATHEMATICS = TO DO A LOT OF EXERCISES YOURSELF, UNDERSTAND MISTAKES AND DO THE EXERCISES AGAIN CORRECTLY