Logarithmic Models
Here are some of the many places logarithms are used:
1. Richter Scale (measurement of intensity of an earthquake)
2. Human Memory Model (what % of information we remember over time.)
3. Sound Intensity (decibels)
4. pH (measure of acidity or alkalinity)
The Richter Scale
The Richter scale measures the magnitude of the seismic waves of an
earthquake (not, as commonly thought, the energy release). An earthquake
measuring “6” on the Richter scale actually has seismic waves 10 times greater
in amplitude of those measuring “5” on the Richter scale, not 1. The equation for
x
Richter magnitude is R(x) = log , where Io is the intensity of a zero-level
Io
earthquake. Many times, Io is said to equal 1.
Example1: If an earthquake is measured having an intensity 100 times that of a
zero-level earthquake, what is its value on the Richter scale?
Example 2: The 2004 Indian Ocean earthquake is estimated to have been about
9.15 on the Richter scale. (The second largest ever recorded!) Find the intensity
of this earthquake if Io is 1.
Human Memory Model
The Human Memory Model approximates the percentage of information the
average person can recall after a certain period of months have passed by. There
are several different formulas given for this, but they all give answers within a few
points of one another. One of the models is as follows:
f(t) = 75 – 6•ln (t + 1), where 0 ≤ t ≤ 12
In this case, t is the number of months that have gone by after being presented
the information (up to one year) and f(t) is the percent of the information retained.
In other words, after “t” months, the average person remembers “f(t)” percent of
the presented information.
Example 3: What percentage was retained 6 months after being presented the
information?
Example 4: After how long did the average person retain only half of the
presented information? Is this answer valid?
Sound Intensity
Sound intensity is measured in something called “decibels”. A decibel,
1
which is
of a Bel and named after Alexander Graham Bell, is measured as
10
watts
.
follows: D(x) = 10 • log 1016 x , where x is the intensity of the sound in
2
cm
Some common decibel levels are listed below.
(
)
Sounds
dB SPL
Rocket Launching
180
Jet Engine
140
Thunderclap, Air Raid Siren 1 Meter
130
Jet takeoff (200 ft)
120
Rock Concert, Discotheque
110
Firecrackers, Subway Train
100
Heavy Truck (15 Meter), City Traffic
90
Alarm Clock (1 Meter), Hair Dryer
80
Noisy Restaurant, Business Office
70
Air Conditioning Unit
60
Light Traffic (50 Meter), Average Home 50
Living Room, Quiet Office
40
Library, Soft Whisper (5 Meter)
30
Broadcasting Studio, Rustling Leaves
20
Hearing Threshold
0
Noise levels above 85 dB will harm hearing over time. Noise levels above 140dB
can damage hearing after just one exposure.
Example 5: Ordinary conversation occurs at a level where the sound intensity is
10 – 10 w/cm2. What decibel level is this? List it in the proper box above.
pH
pH measures the hydrogen ion (written H+ or H30+) concentration in a
substance. The range of the scale of pH is between 0 and 14, where 7 is neutral
(neither acidic or basic). pH values between 0 and 7 are acidic, and values
between 7 and 14 are basic (or alkaline).
Both strong acids and strong bases can “burn”. Stomach acid has a pH of
somewhere between 2 and 3; household ammonia has a pH of about 12. Blood
has a pH of between 7.35 and 7.45; values outside this range can be fatal!
The formula for calculating pH is pH = - log x, where “x” is the hydrogen ion
concentration in moles per liter.
Example 6: What is the pH of a substance that has a hydrogen ion concentration
of 10 – 8.5?
Example 7: What is the “safe” range of the H+ concentration for blood given the
pH values above?