REPRESENTING REAL LIFE SITUATION USING LOGARITHMIC FUNCTION || Applications of Logarithmic Function

Summary

This 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

00:00

🌍 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.

05:04

💥 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.

10:05

🎵 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.

15:07

🧪 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.

Keywords

💡Logarithmic Function

A logarithmic function is a mathematical relationship that is used to express the logarithm of a value. In the video, logarithmic functions are used to model real-life situations like measuring earthquake magnitude, sound intensity, and pH levels. The script illustrates its application in the Richter scale to measure earthquake intensity and in sound decibel levels.

💡Richter Scale

The Richter scale is a logarithmic scale used to measure the magnitude of earthquakes based on the energy released. Charles Richter introduced it in 1935. In the video, the formula for calculating the earthquake magnitude using the Richter scale involves logarithmic functions and energy ratios.

💡Seismology

Seismology is the study of earthquakes and the propagation of elastic waves through the Earth. It plays a crucial role in the video as it explains the relationship between logarithmic functions and the calculation of earthquake magnitudes using the Richter scale.

💡Magnitude

Magnitude in this context refers to the size or energy released by an earthquake. It is calculated using a logarithmic formula. The video explains how the energy released by an earthquake (in joules) is used to determine its magnitude on the Richter scale.

💡Energy (Joules)

Energy, measured in joules, is the amount of energy released during events like earthquakes or sound emissions. In the video, energy values are used to compute the magnitude of earthquakes and the intensity of sound, applying logarithmic functions for comparison with reference values.

💡Decibel (dB)

The decibel is a logarithmic unit used to express the intensity of sound. In the video, the formula for calculating sound intensity in decibels is discussed, showing how a logarithmic function can convert sound energy to a scale that humans can interpret, like the 60 dB level mentioned for normal office noise.

💡Sound Intensity

Sound intensity is the amount of sound energy per unit area, typically measured in watts per meter squared. The video explains how to calculate the sound intensity using logarithmic functions to compare it with the least audible sound level, showing that some sounds are millions of times more intense.

💡pH Level

pH level measures the acidity or alkalinity of a water-based solution. The pH scale is logarithmic, and the video demonstrates how to calculate the pH of a solution by taking the negative logarithm of the hydrogen ion concentration. A pH of 5, for instance, represents a weakly acidic solution.

💡Hydrogen Ion Concentration

Hydrogen ion concentration represents the amount of hydrogen ions in a solution, which determines its acidity. In the video, this concentration is used in a logarithmic formula to calculate pH levels. For example, a concentration of 10^-5 moles per liter corresponds to a pH level of 5.

💡Acidity

Acidity refers to how acidic a solution is, based on the concentration of hydrogen ions. The video shows that solutions with a pH level below 7 are acidic. For example, a solution with a pH of 5 is mildly acidic, and this is calculated using logarithmic functions.

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