Transcript 3.2 – Logarithmic Functions and Their Graphs

Extra 5 point pass if you can
solve (and show how)…
 Find the inverse of:
10𝑙𝑛𝑥
𝑓 𝑥 =
𝑙𝑛𝑥 + 1
*10 minute limit!!!
3.2 – Logarithmic Functions and
Their Graphs
Some things to ponder….
What are the properties of
exponential functions that we
learned yesterday?
Who remembers how to determine
if a function has an inverse?
Will an exponential function have
an inverse?
y = ax
has an inverse logax=y
y = ax
is equivalent to logay=x
Remember that logs are exponents….
So logax is the exponent to which “a”
must be raised to obtain x
 Ex. 1) log28=?
 Ex. 2) log232=?
 Ex 3) log10(1/100)=?
Log4774000=?
x
55 =22500
Graphing Logs…
y=logax
Domain: (0,∞)
Range: (- ∞, ∞ )
x intercept: (1,0)
increasing: (0, ∞)
Graph f(x)=log2x
Graph f(x)=log3x + 4
Transformations…..
 f(x)=logbx
g(x)= alogb(c(x-h))+k
 The transformations are the same for “a”, “c”,
“h”, and “k” for all the other functions we
have studied….*absolute value, quadratic,
exponential, etc.
Natural Log Function…
f(x)=logex
lnx
y=ex and y = lnx are inverses
y=lnx implies
y
e =x
Properties…
 e0=
 e1=
 ln ex=
 elnx=
 ln(1)=
 ln(0)=
 ln(-1)=
 If lnx = lny then
Simplify with out a calculator:
1
𝑒
(a) ln
(b) e
(c)
ln5
𝑙𝑛1
3
(d) 2 lne
Day 1 - HW
pg. 216 #’s 1 – 52 (3’s)
Bacteria in a bottle…
 There is a single bacterium in a bottle at 11:00pm, and it is a type
that doubles once every minute. The bottle will be completely
full of bacteria at 12:00 midnight – exactly one hour.
 In your opinion, what percentage of the bottle will be full when
the bottle starts to look full? For what amount of time between
11:00 and 12:00 would they have plenty of room to grow and
spread out? If you were a researcher in the lab, at what time
between 11:00 and midnight might make you look in the bottle
and think “I’d better get a bigger container for those bacteria!”?
Finding Domain of Ln Functions…
 f(x)=ln(x-2)
 g(x)=ln(2-x)
 h(x)=lnx2
*think about the properties
of ln
Lets do the application (ex 10) on page
215 together…
Graph #41 on page 216
Practice Problems to work on now
pg. 216 #’s 20, 24, 26, 43, 47, 57, 59,
60, 61