MHF 4U3 – ADVANCED FUNCTIONS
Unit 1 – Logarithmic Functions
Lesson #5 – Where We Use Logarithms
Logarithmic Scales:
Logarithmic scales are useful for measuring quantities that can have a very large
range, because logarithms enable us to make large or small numbers more
manageable to work with. Examples of logarithmic scales include the Richter
scale, which measures earthquakes, the pH scale, which measures acidity, and
the decibel scale, which measures sound.
Earthquakes:
The intensities of earthquakes vary over an extremely wide range. To make
such a wide range more manageable, a compressed range, called the Richter
scale, is used. To make the scale convenient, a “standard earthquake,” with a
certain intensity I 0 , is given a magnitude of 0. Earthquakes with intensities
weaker than this standard are so weak that they are hardly ever discussed. Only
magnitudes greater than 0 are used in practice.
I
, where I
I0
is the intensity of the earthquake, and I 0 is the intensity of a standard earthquake.
Thus, a range of earthquake intensities from I 0 to about 8000000000I 0 corresponds
to a range in magnitudes on the Richter scale from 0 to about 8.9.
The magnitude, M , of an earthquake is given by the equation M = log
The equation M = log
M = log
I
can be rearranged:
I0
I
I0
I
= 10 M
I0
I = I 0 × 10 M
From this equation, you can see that, for every increase in the intensity of an
earthquake by a factor of 10, the magnitude on the Richter scale increases by 1.
For example, an earthquake of magnitude 4 is 10 times as intense as an
earthquake of magnitude 3, and 100 times as intense as an earthquake of
magnitude 2.
Example 1: a) On September 26, 2001, an earthquake in North Bay measured
5.0 on the Richter scale. What is the magnitude of an earthquake
3 times as intense as North Bay’s earthquake?
b) On February 10, 2000, Welland experienced an earthquake of
magnitude 2.3 on the Richter scale. On July 22, 2001, St.
Catharines experienced an earthquake of magnitude 1.1. How
many times more intense was the earthquake in Welland?
Solution:
I
,
I0
Let I represent the intensity of North Bay’s earthquake.
I
Then M = 5.0 , and thus, 5.0 = log .
I0
An earthquake three times as intense as North Bay’s
earthquake has an intensity of 3I . So the magnitude of an
earthquake three times as intense as North Bay’s earthquake is:
3I
M = log
I0
a) We use the formula M = log
⎡
I ⎤
M = log ⎢3 × ⎥
⎣ I 0 ⎦
M = log 3 + log
I
I0
M = log 3 + 5.0
M ≅ 5.5
So an earthquake of magnitude 5.5 is approximately three times
as intense as an earthquake of magnitude 5.0.
b) Let M1 = 2.3 represent the magnitude of the earthquake in Welland.
Then I 1 is the intensity of the earthquake in Welland.
Let M 2 = 1.1 represent the magnitude of the earthquake in St. Catharines.
Then I 2 is the intensity of the earthquake in St. Catharines.
I
We want 1 .
I2
I 1 I 0 × 10 M1
=
I 2 I 0 × 10 M 2
I 1 I 0 × 10 M1
=
I 2 I 0 × 10 M 2
I 1 10 2.3
=
I 2 10 1.1
I1
= 10 2.3−1.1
I2
I1
= 101.2
I2
I1
≅ 15.8489
I2
Therefore, the earthquake in Welland was approximately 16
times as intense as the earthquake in St. Catharines.
Chemistry:
The pH scale, which measures the acidity of substances, is another logarithmic
scale. The pH of a solution is a measure of relative acidity in moles per litre,
mol/L, compared with neutral water, which has a pH of 7. If pH < 7, the solution
is classified as acidic and each 1 unit decrease in pH represents a 10-fold
increase in acidity. If pH > 7, the solution is basic or alkaline and each 1 unit
increase in pH represents a 10-fold increase in alkalinity. The pH scale ranges
from 0 to 14.
The pH scale is widely used by chemists, for example, who regularly test the pH of
drinking water to ensure it is safe from contaminants. The pH of a solution can be
represented by the equation pH = − log H + , where H + is the number of moles of
hydrogen ions per litre.
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[ ]
The pH scale makes very small numbers manageable. For example, if
H + = 0.000001 mol/L, pH = 6. The equation pH = − log H + can be rewritten as
H + = 10 − pH . Thus, when the pH level increases by 1, H + is divided by 10. For
1
example, a substance with pH = 4 has
the H+ concentration of a substance
10
1
+
with pH = 3, and
the H concentration of a substance with pH = 2.
100
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Example 2: a) The hydrogen concentration of a sample of water is 6.82 × 10 −8
moles of H+ per litre of water. What is the pH level?
b) A sample of orange juice has a pH level of 3.5. Find its hydrogen
ion concentration.
Solution:
[ ]
a) To solve for pH, we use the formula pH = − log H + and substitute
H + = 6.82 × 10 −8 .
[ ]
[ ]
[
pH = − log H +
pH = − log 6.82 × 10 −8
pH ≅ 7.2
Therefore, the water has a pH level of approximately 7.2, which
means it is slightly basic.
]
[ ]
b) We use the formula H + = 10 − pH and substitute pH = 3.5 to
determine the hydrogen ion concentration of the juice.
H + = 10 − pH
H + = 10 −3.5
H + ≅ 3.16 × 10 −4
Therefore, the hydrogen ion concentration of the orange juice is
approximately 3.16 × 10 −4 mol/L.
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Sound:
The decibel scale, dB, is also a logarithmic scale. The decibel scale measures
sound levels. The human ear can detect a very wide range of sounds, ranging
from a soft whisper to loud machinery. The threshold of pain is about 120 dB.
Examples of other measurement include normal conversation at about 50 dB,
and a jet taking off at about 140 dB. Each increase of 10 dB represents a 10-fold
increase in loudness. For example, the increase from sound of normal
conversation to a jet taking off is 90 dB, so the increase in loudness is 109, or
1 000 000 000. Hence, a jet taking off is 1 000 000 000 times as loud as normal
conversation.
The minimum intensity detectable by the human ear is I 0 = 10 −12 W/m2 (watts per
square metre), and is used as the reference point. The sound level
I
corresponding to an intensity I watts per square metre is L = 10 log .
I0
Example 3: Damage to the ear can occur with sound levels that are greater than
or equal to 85 dB. Find the sound level of a rock concert with an
intensity of 80 W/m2 to determine if fans at the concert are at risk for
hearing damage.
Solution:
We use the formula L = 10 log
I
and substitute I = 80 and I 0 = 10 −12 .
I0
I
I0
80
L = 10 log −12
10
L ≅ 139
Therefore, the concert has a sound level of 139 dB, which means
that people attending the concert may be at risk for hearing damage.
L = 10 log
Homework:
pg. 359 #8-11, pg. 376 #13-16, pg. 385 #11,
pg. 392 #8, 10