1. 9.3 Logarithmic Functions (Day 2)

2. Recall the graphs of y = logb x:
f (x) = logb x has the y-axis as its asymptote
The y-axis is x = 0, so if we change the function to f (x) = logb (ax + c),
then ax + c = 0 is the asymptote.
The x-int is found by letting y = 0 & y-int is when x = 0
b > 1
0 < b < 1

3. Recall the graphs of y = logb x:
f (x) = logb x has the y-axis as its asymptote
The y-axis is x = 0, so if we change the function to f (x) = logb (ax + c),
then ax + c = 0 is the asymptote.
The x-int is found by letting y = 0 & y-int is when x = 0
Ex 1) Find the asymptote and x- & y-int of h(x) = log (3x + 10)
asymptote:
3x + 10 = 0
3x = –10
b > 1
0 < b < 1
x-int:
0 = log (3x + 10)
100 = 3x + 10
1 = 3x + 10
–9 = 3x
x = –3
y-int:
y = log (3(0) + 10)
y = log (10)
10y = 101
y = 1
(0, 1)
(–3, 0)

4. We can use our calculators to estimate logs  we will round to a typical 4 decimal places.
Ex 2) Which is wrong?
log 510 = b) ln 70.5 = c) log 0.03 =
Change of Base Formula:
typically:
which is
written as
(used in calculus a lot!)
Ex 3) Calculate log2 7
Half Class: Half Class:
both

5. Logarithms are used to model a wide variety of problems.
For example, magnitude of sound, Decibels, is
where I = intensity of sound & I0 is intensity
of the threshold of hearing, which is 10–16 W/cm2
Ex 4) Determine the loudness of each sound to the nearest decibel.
a) Whisper: I = 3.16 × 10–15
b) A subway train: I = 5.01 × 10–7
Another application is the Richter Scale to measure the magnitude of earthquakes.
x0 is magnitude of zero-level earthquake w/ reading of mm
x is seismographic reading of quake

6. Ex 5) Determine the equation of the function pictured below
Ex 5) Determine the equation of the function pictured below. It is a logarithmic function that has been translated.
Answer will be of this form:
Use the asymptote to find c
Now use a point on the graph (NOT the point on the x-axis) to solve for b – Use (1, 2)