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 FunctionsWhy 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 bc = xlog2(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…
(1) log10 = 1
(2) lne = 1
Practice
Evaluate, if possible. If not, state so.
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
Can you state any characteristics?–Asymptotes, x - intercepts, Domain, Range
• Graph of y = logb(x) when 0<b<1
Practice
Graph. State the domain and range.
Change of Base Formula
This formula allows us to compute logs using the calculator, by converting to base 10 or e.
Example: