Transcript PCH (4.5) Props. of Logs 11

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 time
Now, it’s time for Something Completely
Different (any Monty Python fans in the
room?)
Formative vs. summative assessments
The Good Parts Version…