Logarithmic Functions
• Categorized as Transcendental (Non-Algebraic)
Functions
• Inverse of an Exponential Function
• Many real-life situations can be modeled using
logarithmic functions, including:
– Earthquake Intensity
– Human Memory
– Pace of life in Large Cities
Definition of Logarithmic Function
For x > 0 and a > 0, a≠ 1,
𝑦 = 𝑙𝑜𝑔𝑎 x if and only if x = ay
This function, f(x) = 𝑙𝑜𝑔𝑎 x is called the
logarithmic function with base a.
Essentially, a logarithm is an exponent.
Examples
Logarithmic Form
𝑙𝑜𝑔2 8
𝑙𝑜𝑔5 5
𝑙𝑜𝑔4
1
16
Exponential Form
2x = 8
Two to what exponent is 8?
5x = 5
Five to what exponent is 5?
4x
=
1
16
Four to what exponent is
1
?
16
Examples
Logarithmic Form
𝑙𝑜𝑔10 100
We call “log” the “common
log”
𝑙𝑜𝑔𝑒 1
We call “ln” the “natural log”
Exponential Form
10x = 100
10 to what exponent is 100?
We call “10” the “common
base”
ex = 1
e to what exponent is 1?
We call “e” the “natural
base”
Properties of Logarithms
• 𝑙𝑜𝑔𝑎 1 = 0
• 𝑙𝑜𝑔𝑎 𝑎 𝑥 = x
• 𝑙𝑜𝑔𝑎 a=1
• 𝑎𝑙𝑜𝑔𝑎 𝑥 = x
• If 𝑙𝑜𝑔𝑎 x = 𝑙𝑜𝑔𝑎 y, then
x= y
Logarithmic Function
f(x) = 𝑙𝑜𝑔𝑎 𝑥
• Domain (0, ∞)
• Range (-∞, ∞)
• No y-intercept since y-axis
is vertical asymptote
• x-intercept (1, 0)
• Continuous
• Neither even nor odd
• Increasing (0, ∞)
• No relative minimum or
maximum
• Has an inverse
(exponential)
X
1
2
4
8
y=f(x)
𝑓 𝑥 = 𝑑 + 𝑙𝑜𝑔𝑎 (bx-c)
•
•
•
•
•
•
•
Transformations learned in Chapter 1 still apply
Parent is exponential function with base a
Vertical translation –”d”
Horizontal translation –”bx-c=0”
Reflection on x-axis – “sign of a”
Reflection on y-axis-”sign of b”
Vertical Stretch or Shrink – “numeric value of a”
• EXAMPLES
Applications
• The function 𝑓 𝑥 = 13.4 + 46.3 ln 𝑥 models
the percentage of inbound email in the United
States considered “spam”, f(x), where x is the
number of years after 2000. According to this
model, what percentage of email was
considered spam in 2003 (round to nearest
percent)?
Applications
• The percentage of adult height attained by a
girl who is x years old can be modeled by
f(x) = 62 +35log (x -4)
where x represents the girl’s age (from 5-15)
and f(x) represents the percentage of her
adult height. According to the model, what
percentage of her adult height has a girl
attained at age 13, rounded to nearest tenth?
Applications
• The loudness, L(x), measured in decibels, of a
sound of intensity, x, measured in watts per
square meter is defined as
𝐿 𝑥 =
𝑥
10𝑙𝑜𝑔 −12
10
Determine the loudness, in decibels, of
normal conversation intensity of x = 10-7
watt per square meter
Assignment
• Page 411 #53, 55, 59-63 odd, 113, 115, 119