REPRESENTING REAL LIFE SITUATION USING LOGARITHMIC FUNCTION || Applications of Logarithmic Function
Summary
TLDRThis tutorial explores real-life applications of logarithmic functions, specifically in seismology, acoustics, and pH levels. It explains how logarithms are used to calculate earthquake magnitudes on the Richter scale, sound intensity in decibels, and the pH levels of solutions. Using detailed examples, the video shows how to apply logarithmic functions to determine the magnitude of an earthquake, the intensity of sound, and the acidity of a solution, while highlighting their significance in understanding natural phenomena. The tutorial encourages viewers to engage with problems and share their solutions.
Takeaways
- 🌍 The Richter scale is used in seismology to measure earthquake magnitude using logarithmic functions.
- 📈 The magnitude of an earthquake is based on the logarithm of the ratio of energy released compared to a reference earthquake.
- 🧮 The formula for earthquake magnitude is: R = (2/3) * log(E / 10^4.40), where E is the energy released by the earthquake.
- 💥 In the example, an earthquake releasing 10^12 joules of energy has a magnitude of 5.1 on the Richter scale, indicating it is a strong earthquake.
- 🔍 The earthquake releases approximately 39,810,717 times more energy than the reference earthquake.
- 🔊 In acoustics, the decibel level of sound is calculated using the formula D = 10 * log(I / 10^-12), where I is the sound intensity.
- 📏 For a sound intensity of 10^-6 watts per meter squared, the sound level is 60 decibels, which is considered a normal sound level.
- 🔊 The sound in a quiet office (10^-6 watts per meter squared) is 1 million times more intense than the least audible sound.
- 🧪 The pH level of a solution is determined using the formula pH = -log([H+]), where [H+] is the concentration of hydrogen ions in moles per liter.
- ⚗️ A solution with a pH of 5 is considered acidic since it's below 7 on the pH scale, indicating a higher concentration of hydrogen ions.
Q & A
What is the significance of using a logarithmic function to measure earthquake magnitude?
-A logarithmic function is used to measure earthquake magnitude because it allows the representation of large variations in energy release in a more manageable scale. This makes it easier to quantify and compare the intensity of earthquakes.
How is earthquake magnitude calculated using the given formula?
-The earthquake magnitude (R) is calculated using the formula R = 2/3 log (E / 10^4.4), where E is the energy released by the earthquake in joules. This formula compares the energy of the earthquake to a reference earthquake, which releases 10^4.4 joules of energy.
In the example provided, how much energy does the earthquake release, and what is its magnitude?
-The earthquake releases approximately 10^12 joules of energy. Using the formula, its magnitude on the Richter scale is calculated as 5.1.
How much more energy does the earthquake release compared to the reference earthquake?
-The earthquake releases approximately 39,810,717 times more energy than the reference earthquake.
What is the decibel level formula used to calculate sound intensity?
-The decibel level (D) of sound is calculated using the formula D = 10 log (I / 10^-12), where I is the sound intensity in watts per meter squared, and 10^-12 watts per meter squared is the least audible sound a human can hear.
How is the sound intensity in decibels calculated for a quiet office with a sound intensity of 10^-6 watts per meter squared?
-For a sound intensity of 10^-6 watts per meter squared, the decibel level is calculated as 60 decibels using the formula D = 10 log (10^-6 / 10^-12).
How much more intense is the sound in the quiet office compared to the least audible sound?
-The sound in the quiet office is 1 million times more intense than the least audible sound a human can hear, as represented by 10^6.
What is the pH formula used to measure acidity in water-based solutions?
-The pH of a water-based solution is calculated using the formula pH = -log [H+], where [H+] is the concentration of hydrogen ions in moles per liter.
In the example given, what is the pH level of a solution with a hydrogen ion concentration of 10^-5 moles per liter?
-The pH level of the solution is calculated as 5, which indicates the solution is acidic since it is below the neutral pH of 7.
Why is it important to use logarithmic functions in measuring sound intensity, earthquake magnitude, and pH levels?
-Logarithmic functions allow for the compression of large ranges of values, making it easier to interpret, compare, and analyze phenomena that involve exponential changes, such as sound intensity, earthquake magnitude, and acidity.
Outlines
🌍 Understanding Earthquake Magnitude Using Logarithmic Functions
This paragraph introduces the application of logarithmic functions in seismology to measure earthquake magnitude. The Richter scale, proposed by Charles Richter in 1935, uses a logarithmic scale to represent the magnitude of an earthquake based on the energy it releases. The formula R = \frac{2}{3} \log\left(\frac{E}{10^{4.40}}\right) calculates the magnitude, where E is the energy in joules released by the earthquake and 10^{4.40} joules is the energy from a small reference earthquake. A worked-out example shows how to compute the magnitude of an earthquake that releases 10^{12} joules of energy, resulting in a magnitude of approximately 5.1 on the Richter scale.
💥 Comparing Earthquake Energy Release
This paragraph continues the discussion on earthquake magnitude by explaining how much more energy an earthquake releases compared to a reference earthquake. The ratio of the energy released by an earthquake to that of a reference earthquake is used to measure this difference. In the example, the earthquake releasing 10^{12} joules of energy produces 39.8 million times more energy than the reference earthquake. The paragraph concludes with the transition to sound intensity and how logarithmic functions are also used to measure decibel levels in acoustics.
🎵 Measuring Sound Intensity with Decibels
This section introduces the concept of sound intensity using decibel levels, which also employs logarithmic functions. The decibel level D = 10 \log\left(\frac{I}{10^{-12}}\right) is calculated, where I represents the sound intensity in watts per meter squared and 10^{-12} is the least audible sound a human can hear. An example is provided where the sound intensity in a quiet office is 10^{-6} watts per meter squared, corresponding to a decibel level of 60 dB, which is considered non-hazardous. The section concludes by calculating that the sound in this example is 1 million times more intense than the least audible sound.
🧪 Exploring Acidity and pH Levels
This paragraph discusses the use of logarithmic functions to calculate the pH levels of solutions, a measure of acidity. The formula pH = -\log\left([H^+]\right) calculates the pH level based on the concentration of hydrogen ions in moles per liter. An example shows a solution with 10^{-5} moles per liter of hydrogen ions, leading to a pH level of 5, indicating that the solution is acidic. The paragraph emphasizes how logarithmic functions are essential for determining the pH levels of various solutions.
👋 Conclusion and Closing Remarks
The final paragraph closes the video with a brief remark on the concept of inequalities. It transitions away from logarithmic functions, signaling the end of the lesson with a musical outro and a farewell message.
Mindmap
Keywords
💡Logarithmic Function
💡Richter Scale
💡Seismology
💡Magnitude
💡Energy (Joules)
💡Decibel (dB)
💡Sound Intensity
💡pH Level
💡Hydrogen Ion Concentration
💡Acidity
Highlights
Introduction to real-life applications of logarithmic functions in seismology and sound intensity.
Explanation of the Richter scale for measuring earthquake magnitude using logarithmic functions.
Detailed formula for calculating earthquake magnitude: R = (2/3) log(E / 10^4.40), where E is energy in joules.
Example of calculating the magnitude of an earthquake that releases 10^12 joules of energy, resulting in a magnitude of 5.1 on the Richter scale.
Clarification that an earthquake with a magnitude of 5.1 on the Richter scale is classified as 'strong.'
Comparison of energy released by the earthquake to that of a reference earthquake, showing it releases 39.8 million times more energy.
Transition to sound intensity and decibel level calculation using the logarithmic function: D = 10 log(I / 10^-12), where I is sound intensity in watts per meter squared.
Example of calculating the decibel level of sound in a quiet office with intensity of 10^-6 watts per meter squared, resulting in 60 decibels.
Clarification that 60 decibels is considered a non-hazardous, normal sound level.
Demonstration of how the sound is one million times more intense than the least audible sound a human can hear.
Introduction to logarithmic function applications in acidity and pH level calculations.
Definition of pH as the negative logarithm of the concentration of hydrogen ions in a water-based solution.
Example of calculating pH level for a solution with 10^-5 moles of hydrogen ions, resulting in a pH of 5.
Explanation that a solution with a pH of 5 is classified as acidic, based on the pH scale.
Conclusion and encouragement for viewers to solve additional problems related to inequalities and logarithmic functions.
Transcripts
[Music]
good day everyone
welcome back samat usai general match
tutorial
real life application logarithmic
function let's begin
[Music]
ilanza mangapinaka popular application
non logarithmic function i suffield nang
seismology
represent a real life situation gamet
and logarithmic
function
una we have the earthquake magnitude on
a register
scale so paano baginagamit a concept on
a logarithmic
function identifying magnitude
is an earthquake in 1935
charles richter proposed a logarithmic
scale to measure the intensity of an
earthquake he defined the magnitude of
an earthquake as a function
of its amplitude on a standard
seismograph
so young formula identifying a magnitude
is an earthquake is given by this
function
magnitude r is equal to two third
log of e all over
ten raised to four point
four t now
is in joules is the energy released
by the earthquakes
raised to 4.40 joules
is the energy released by a very small
reference earthquake so the formula
indicates
that your magnitude of an earthquake is
based on
the logarithm of the ratio between the
energy it releases
and the energy released by reference
earthquake again para malama new
magnitude is an
earthquake in a gamete nothing
logarithmic function
to third log of e all over 10
raised to 4.40 so capacitor nothing
magnitude is an earthquake but nothing
let's try to represent problem
that involves calculating the
the magnitude of an earthquake
example number one suppose
an earthquake released approximately 10
raised to 12
joules of energy so
is what is its magnitude on a richter
scale and in pangala how much more
energy does this earthquake release
than the reference earthquake so
to represent this using
logarithmic function alumni as a letter
a then we want to solve the magnitude of
the earthquake
energy released by an earthquake is 10
raised to 12
joules right gamition formula
r is equal to two third log of e
all over 10 raised to 4.40
is a substitute nothing new value
e so when we plug in the volume e we
have
2 third log 10 raised to 12
all over 10 raised to 4.40
so simplifying this we have
two third log of ten
raised to seven point six bucket taking
seven point six
numerator at denominator so debunking
nagawa nothing is we subtract
its exponent parameter simplify so
12 minus 4.40 that gives you
seven point six k negan then raised to
7.6
yen log
of 10 raised to 7.6 i equivalent lungs
are 7.6
so 2 3 times 7.6
the magnitude is approximately equal to
5.1 so gammitian reached third scale
by the nothing mcclassify young
earthquake as
strong casinga
so we can classify the earthquake as
strong
how about letter b antenna how much more
energy does this
earthquake release than the reference
earthquake new
acting formula indicates that your
magnitude of an earthquake is based on
the logarithm of the ratio between
energy released by an earthquake
and the energy released by a reference
earthquake
so this earthquake releases
39 million eight hundred ten thousand
seven hundred
seventeen times more energy
than the reference earthquake
nothing a reference earthquake
we divide 10 raised to 12
by 10 raised to 4.40
kasingha in 10 raised to 4.40 is your
reference
earthquake
earthquake reference earthquake
[Music]
claro
this time pagusa panama natin
u sound intensity
in acoustic the decibel
level of sound is given by
this function d is equal to
10 log of i
all over 10 raised to negative
12. where jung ainaten is the sound
intensity in watts
per meter squared and new 10 raised to
negative 12 watts per meter squared
is the least audible sound a human
can hear so once in a calculated nothing
new sound intensity
but nothing may identify co-annoying
level nang sound
so let's have an example
nothing you represent and sound
intensity
gametang logarithmic function
example number two the decibel level of
a sound
in a quiet office is 10 raised to
negative 6
watts per meter squared
what is the corresponding sound
intensity in decibel
balantano how much more intense
is this sound than the least audible
sound a human can hear
first let's calculate
young sound intensity ai natin is equal
to 10
raised to negative six watts
per meter squared so gamma
in formula and sound intensity so
d is equal to 10 log of i
all over 10 raised to negative 12 if i
plug in nothing
value num i so we have
10 log of 10
raised to negative 6 all over 10
raised to negative 12.
now we simplify this we have 10
log of 10 raised to 6 bucket man
take note that 10 raised to negative 6
divided by ten raised to negative twelve
and
yari that like they divide dial up the
same base
you copy the base and subtract the
exponent
so negative six minus the negative
twelve that gives
you positive six
now from here we simplify
ten log of ten
raised to six young volume log
10 raised to 6 i 6
so 10 times 6 that gives you
the sound intensity is equal to
60 decibel so gametoting
sound level chart so you 60
decibel is considered non hazardous
sound sound level
normal
now how about young letter b
how much more intense is this sound
than the least audible sound a human can
hear
the bangla young least audible sound a
human can hear
is 10 raised to negative 12
watts per meter squared right
we have the sound is actually
one million times more intensity
than the least audible sound a human can
hear kasingan
how much more intense is the sound than
the least audible
sound a human can hear so they divide
nothing and 10 raised to negative 6
by 10 raised to negative
12. so again that is equivalent to 10
raised to 6
and 10 raised to 6 is 1 million
kappa pinago sapan and sound intensity
subway nothing a represent some problem
gamete logarithmic function so
since unsung intensity i made for milan
so tanda and formula not some intensity
i
d is equal to 10 log of i
all over 10 raised to negative 12.
claro manateen
it represents human problems involving
acidity
and ph level gametang logarithmic
function so
capalpina goes up and in acidity and ph
level the ph level of a water-based
solution
is defined by the given function
ph is equal to the negative log
of the concentration of hydrogen ions
in moles per liter
is actually the concentration of
hydrogen ions
in moles per liter
capacitor computation and ph level
non-water-based solution
then there are nothing at all solutions
with a ph
of seven are defined neutral
samantha among a ph level
seven i considered acidity
and those with ph level not greater than
seven
i think nothing basic
let's have an example represent
ang problems involving acidity and ph
level gamete and logarithmic
function example number three
a one liter solution contains zero
0.00001
moles of hydrogen ions
find its ph level
since there are 0.0001 moles of hydrogen
ions in one liter
then your concentration of hydrogen ions
is
ten raised to negative five moles per
liter
so the marinating volume
concentration of hydrogen ions
so we will just plug this one in our
formula now ph level
so in formula atom is ph is equal to
negative log
times the concentration of hydrogen ions
in moles per liter
so in volume concentration of hydrogen
ions
is 10 raised to negative
5 moles per liter so we plug in that one
so we have negative log
of 10 raised to negative 5.
take note the log of 10
raised to negative 5 is equivalent to
negative 5 so we have negative times the
negative 5
so your ph level is 5.
looking at the ph scale so young
solution nothing with
ph of 5 are considered
a cd debug has less than 7
mph level so this is how we represent
problems involving acidity and ph level
gamma logarithmic function
iph level in formula or young adding
function i
ph is equal to negative log
of the concentration of hydrogen
ions claro
please pause the video and try to solve
each of the following problem
share your answers in the comment
section
[Music]
is
inequalities see ya bye
you
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