2. Logarithms are essential in our day to day
living. Its application into the real world is
innumerable. To name some, it is used in
determining the intensity of earthquakes
and sounds, calculating compound interests,
computing population growth and decay,
measuring pH levels, and carbon dating.
Logarithmic function
Today's Lesson:
3. Objectives
• define logarithmic
functions
• . show illustrations of
logarithmic functions
that represent real-life
situations
• represent real-life
situations using the
logarithmic functions
4. LOGARITHM
The inverse function is obtained by
interchanging the roles of x and y.
Thus, the inverse of the exponential
function 𝒚 = 𝒃𝒙 is 𝒙 = 𝒃𝒚
The equation 𝒙 = 𝒃𝒚
tells us that y is the
exponent to which b must be raised to produce x
(y is the exponent on b that produces x).
We use the term logarithm to replace the term
exponent.
A logarithm
then is an
exponent.
5. LOGARITHM
We can now say that in the equation 𝒙 = 𝒃𝒚,
𝑦 is the logarithm to which b must be raised to
produce x.
In the words, y is the logarithm on b that
produces x. We write 𝒚 = 𝒍𝒐𝒈𝒃 𝒙 ( read as “ y
equals the logarithm of x to the base b”)
9. KINDS oF LOGARITHM
Common logarithms are logarithms with base 10,
the base is usually omitted when writing common
logarithms.
This means that 101
= 10 is written as 𝑙𝑜𝑔10 = 1
and 102
= 100 is written as 𝑙𝑜𝑔100 = 2 and so on.
Natural logarithms are logarithms with base 𝒆
(which is approximately 2.71828 as mentioned in
the previous module). This means that 𝒍𝒐𝒈𝒆 𝒙 can
be written as ln 𝑥.
19. Richter Scale
The Richter magnitude scale was
developed in 1935 by Charles F.
Richter of the California Institute
of Technology as a mathematical
device to compare the size of
earthquakes.
The magnitude of an earthquake
is determined from the logarithm
of the amplitude of waves
recorded by seismographs.
20. Richter Scale
Earthquake Magnitude on a Richter scale
The magnitude R of an earthquake is given by:
𝑹 =
𝟐
𝟑
𝒍𝒐𝒈
𝑬
𝟏𝟎𝟒.𝟒𝟎
where E (in joules) is the energy released by the earthquake (the
quantity 𝟏𝟎𝟒.𝟒𝟎
is the energy released by a very small reference
earthquake)
The formula indicates that the magnitude of an earthquake is
based on the logarithm of the ratio between the energy it releases,
and the energy released by the reference earthquake.
21.
22. EXAMPLE:
Suppose that an earthquake released approximately
𝟏𝟎𝟏𝟐
joules of energy.
(a) What is its magnitude?
(b) How much more energy does this earthquake
release than by the reference earthquake?
23. EXAMPLE:
Suppose that an earthquake released approximately
1012
joules of energy.
(a) What is its magnitude?
(b) How much more energy does this earthquake release than by the
reference earthquake?
(b)
1012
104.40 = 107.6
≈ 𝟑𝟗𝟖𝟏𝟎𝟕𝟏𝟕
The earthquake released
39810717 times more energy than
the reference earthquake.
(a) 𝑅 =
2
3
𝑙𝑜𝑔
𝐸
104.40
𝑅 =
2
3
𝑙𝑜𝑔
1012
104.40
𝑹 ≈ 𝟓. 𝟏
Magnitude 5 is
described as STRONG
24. SOUND INTENSITY
In acoustics, the decibel (dB)
level of a sound is
where 𝐼 is the sound intensity
in 𝑤𝑎𝑡𝑡𝑠/𝑚2
(the quantity 10−12
𝑤𝑎𝑡𝑡𝑠/𝑚2
(is the least audible
sound a human can hear.
25.
26. EXAMPLE:
The decibel level of sound in an office is
𝟏𝟎−𝟔
𝒘𝒂𝒕𝒕𝒔/𝒎𝟐
.
(a) What is the corresponding sound intensity in decibels?
(b) How much more intense is this sound than the least
audible sound a human can hear?
27. EXAMPLE:
The decibel level of sound in an office is
10−6
𝑤𝑎𝑡𝑡𝑠/𝑚2
.
(a) What is the corresponding sound intensity in decibels?
(b) How much more intense is this sound than the least audible sound a human can
hear?
(b)
𝟏𝟎−𝟔
𝟏𝟎−𝟏𝟐 = 𝟏𝟎𝟔
≈ 𝟏𝟎𝟎, 𝟎𝟎𝟎
The sound is 100, 000 times more
intense than the least audible
sound a human can hear.
(a) 𝐷 = 10 𝑙𝑜𝑔
𝐼
10−12
𝐷 = 10 𝑙𝑜𝑔
10−6
10−12
𝑫 = 𝟔𝟎 𝒅𝑩
60-85 dB is described as
INTRUSIVE.
28. pH Scale
Acidic and basic are two extremes
that describe a chemical property.
Mixing acids and bases can cancel
out or neutralize their extreme
effects.
A substance that is neither acidic nor
basic is neutral.
29. The pH scale measures how
acidic or basic a substance is.
The pH scale ranges from 0 to 14.
A pH of 7 is neutral.
A pH less than 7 is acidic.
A pH greater than 7 is basic.
pH Scale
The pH level of a
water-based solution
is defined as
𝒑𝑯 = − 𝐥𝐨𝐠 𝑯+
where 𝑯+
is the concentration of
hydrogen ions in moles per liter.
30. EXAMPLE:
A 1-liter solution contains 0.01 moles of hydrogen
ions. Determine and describe its pH level
So, 𝑝𝐻 = −(−2) = 2, therefore, the pH level is 2
Since the pH level is 2, then it is acidic.
32. 1. Suppose that an earthquake released approximately
108
joules of energy.
a. What is the magnitude on a Richter scale?
b. How much more energy does this earthquake
release than the reference earthquake?
ACTIVITY 2
2. The intensity of sound of a lawn mower is 10−3
watts/
𝑚2
.
a. What is the corresponding sound intensity in
decibels?
b. How much more intense is this sound than the least
audible sound a human can hear?
