Transcript 4_Chapter_5_files/5.3 notes

Sections 5.3-5.5
Logarithmic Functions (5.3)
What is a logarithm???
LOGS ARE POWERS!!!!
A logarithm or “log” of a number of a certain base is the
exponent to which the base of the log must be raised in
order to produce the number. The base cannot equal 1
and must be greater than 0.
For instance, if logb(x) = c and b≠1 and b>0, then c is the
specific exponent to which you must raise b in order to get
x: bc = x
Logarithmic Functions
Why do we need logs? Let’s explore…
32 = 9 and 33 = 27
but what would we need to raise 3 to in order to get 20??
3a = 20
that’s what logs tell us!!
a = log320
Which two integers is log320 between?
2 and 3
Logarithmic Functions
From the definition, we have stated that if logb(x) = c,
then bc = x under the conditions that b≠1 and b>0.
Why do we need to place any restrictions on b or x so
this can make sense? Let’s try some values…
logb(x) = c, so
c
b =
x
log2(8) = c so 2c = 8
c = 3, so far we are ok
log1(5) = c so 1c = 5
Does not exist; 1c always equals 1
log-2(8) = c so (-2)c = 8
Does not exist; if c = 3, then (-2)3 = -8
log3(-9) = c so 3c = -9
Does not exist; 3c cannot be negative
log2(0) = c so 2c = 0
Does not exist; 2c cannot equal 0
Summary: b≠1, b>0 and x>0
Log Properties
(1)
logbb = 1
(2)
logb1 = 0
common log has base 10: log(x) = log10(x)
natural log has base e: ln(x) = loge(x)
Therefore…
log10 = 1
lne = 1
Practice
Evaluate, if possible. If not, state so.
1)log 5 (125)
2)log 32 (2)
3)log 4 (-16)
4)log 1 (27)
3
y = logb(x - h) + k
When graphing logs we first need to identify
and graph the asymptote. Earlier we discovered
that the argument inside the log must be
greater than 0.
Therefore, x > h so the domain is (h, +∞) and
there must be an asymptote at x = h
The range is all real numbers
Now find three points; the simplest values are
when x - h = 1 and when x - h = b
Graph of a Logarithmic Function
Graph of y = logb(x) when b>1 • Graph of y = logb(x) when 0<b<1
Can you state any characteristics?
–Asymptotes, x - intercepts, Domain, Range
Practice
Graph. State the domain and range.
11)y = log4 x
12)y = log 1 x - 3
2
13)y = log 5 (x - 2)
14)y = log 2 (x + 5) -1
Change of Base Formula
log a c
log b c =
log a b
This formula allows us to compute logs using the
calculator, by converting to base 10 or e.
Example:
log(5) ln(5)
log 3 5 =
=
log(3) ln(3)