Properties of Logs (4.5)

What to do with logs (besides building a fire)

A little POD for the fun of it

How long will it take a population to double at 4.5% annual growth, compounded continuously?

A little POD for the fun of it

How long will it take a population to double at 4.5% annual growth, compounded continuously?

Using the Rule of 72, we could roughly estimate this time: 72/4.5 = 16 years.

Exactly, it is (ln 2)/.045 = 15.4 years.

Laws of Logs

1. loga(cd) = logac + logad

2. loga(c/d) = logac – logad

3. loga(cd)= d logac

What are the possible values for a, c, and d?

Laws of Logs

1. loga(cd) = logac + logad

2. loga(c/d) = logac – logad

3. loga(cd)= d logac

The restrictions for a match the restrictions for an exponential base– positive numbers not equal to 1. (Using the change of base, can we divide by log 1?)

Because we can take the log only of a positive number, c and d must be greater than 0.

Laws of Logs

Let’s look at the proof of the first law: loga(cd) = logac + logad

Let r = logac and s = logad.

Then ar = c and as = d.

cd = (ar)(as)

cd = a (r+s)

logacd = r+s

logacd = logac + logad

Laws of Logs

Could you do something similar for the proof of the second law: loga(c/d) = logac – logad?

Let r = logac and s = logad.

Then ar = c and as = d.

c/d =

Two questions

Given these laws, what might be the law for

loga(c+d)?

loga (c-d)?

(Major foot stomp here.)

Use them

Solve the equation:

log2x + log2(x+2) = 3

Be sure and check your answers!

Use them

Solve the equation:

log2x + log2(x+2) = 3

log2(x(x+2)) = 3 23 = x(x+2)

x2 + 2x – 8 = 0 (x + 4)(x – 2) = 0

x = -4, x = 2, but only one works– test and see

What do you know about negative values of x? Why can’t you take the log of a negative number?

Use them

Solve the equation:

ln(x+6) - ln10 = ln (x-1) - ln2

Use them

Solve the equation:

ln(x+6) - ln10 = ln (x-1) - ln2

ln ((x+6)/10) = ln ((x-1)/2)

(x+6)/10 = (x-1)/2

2(x+6) = 10(x-1)

2x +12 = 10x -10

22 = 8x

x = 11/4

We’ll finish this lesson next timeNow, it’s time for Something Completely

Different (any Monty Python fans in the room?)

Formative vs. summative assessmentsThe Good Parts Version…