BASIC PROPERTIES OF LOGARITHMS || GRADE 11 GENERAL MATHEMATICS Q1

Summary

This educational video script focuses on the fundamentals of logarithms, aiming to enhance understanding and problem-solving skills. It introduces basic properties of logarithms, such as the logarithm of one being zero and the logarithm of a base raised to a power being equal to that power. Practical examples, like calculating decibel levels at concerts and hydrogen ion concentration in vinegar, are used to demonstrate real-life applications of logarithms. The script concludes with a set of five practice problems to reinforce learning, encouraging viewers to apply the concepts discussed.

Takeaways

• 📘 The video focuses on the application of basic properties of logarithms to solve logarithmic equations.
• 🔢 Logarithm of one with any base \( b > 0 \) and \( b \neq 1 \) is zero, expressed as \( \log_b(1) = 0 \).
• 🆗 The logarithm of \( b^x \) with base \( b \) is equal to \( x \), or \( \log_b(b^x) = x \).
• ⬆️ If \( x > 0 \), then \( b \) raised to the logarithm of \( x \) with base \( b \) equals \( x \), or \( b^{\log_b(x)} = x \).
• 🔍 The video provides examples to illustrate the properties, such as calculating \( \log_{10}(10) = 1 \) and \( \log_4(64) = 3 \).
• 🎶 It applies logarithms to real-life scenarios, like calculating the decibel level of a concert with a sound intensity of \( 10^{-2} \) watts per square meter, resulting in 100 decibels.
• 🍯 The video also demonstrates how to calculate the hydrogen ion concentration of vinegar with a pH level of 3.0, which is \( 10^{-3.0} \) moles per liter.
• 📚 The presenter encourages viewers to apply these basic properties to solve five logarithmic expression problems presented at the end of the video.
• 👍 The video concludes with a prompt for viewers to like, subscribe, and hit the bell button for more educational content.
• 📈 The script serves as a tutorial for understanding logarithms, emphasizing their practical applications in various contexts.

Q & A

What are the basic properties of logarithms mentioned in the script?

The script mentions three basic properties of logarithms: 1) The logarithm of one with any base b (where b > 0 and b ≠ 1) is equal to zero. 2) The logarithm of b^x with base b is equal to x. 3) If x > 0, then b raised to the logarithm of x with base b is equal to x.

How is the logarithm of 64 with base 4 calculated in the script?

The script calculates the logarithm of 64 with base 4 by recognizing that 64 is 4 cubed (4 * 4 * 4), which means 4^3 = 64. Therefore, the logarithm of 64 with base 4 is 3.

What is the decibel level of a concert with a sound intensity of 10^-2 watts per square meter according to the script?

Using the formula 10 * log10(I/I0) where I0 is 10^-12 watts per square meter, the script calculates the decibel level to be 10 * (-2 - (-12)) = 10 * 10 = 100 decibels.

How does the script determine the hydrogen ion concentration of vinegar with a pH level of 3.0?

The script uses the formula pH = -log10[H+] to determine the hydrogen ion concentration. Substituting pH = 3.0, the script calculates [H+] = 10^-3.0, which means the hydrogen ion concentration is 10^-3.0 moles per liter.

What is the significance of the property that the logarithm of one is zero in logarithmic calculations?

The property that the logarithm of one is zero simplifies calculations by allowing any logarithm with a base raised to the power of zero to be directly equated to zero, which is a fundamental aspect of logarithmic identities.

Can you explain the concept of 'base' in logarithms as presented in the script?

In the script, the 'base' of a logarithm refers to the number that is raised to the power indicated by the logarithm. For example, in log_b(x), 'b' is the base, and it is the number that must be raised to the power of the logarithm's result to get 'x'.

How does the script use logarithmic properties to solve real-life problems like calculating decibel levels?

The script demonstrates the use of logarithmic properties by applying the formula for decibel calculation, which involves logarithms. It shows how to use the properties of logarithms to simplify the calculation and find the sound intensity level in decibels.

What is the role of the property that b^(log_b(x)) = x in the script's explanation of logarithms?

This property is crucial as it demonstrates the inverse relationship between exponentiation and logarithms. It is used in the script to show how to revert from a logarithmic form back to its original exponential form, which is essential for solving certain types of logarithmic equations.

Why is it important to know that the base of a logarithm must be greater than zero and not equal to one?

The script emphasizes that the base of a logarithm must be greater than zero and not equal to one because these conditions ensure that the logarithm is defined and has real number solutions. A base of zero or one would lead to undefined or infinite values, which are not useful in most mathematical applications.

How does the script use logarithms to find the value of complex logarithmic expressions?

The script uses the basic properties of logarithms to simplify complex expressions. It demonstrates how to break down expressions using properties like log_b(b^x) = x and log_b(1) = 0, and then combines these to find the values of more complicated logarithmic expressions.

Outlines

00:00

📘 Introduction to Logarithms

This paragraph introduces the basic properties of logarithms with real numbers b and x, where b is greater than zero and not equal to one. The properties discussed include: (1) log_b(1) = 0, (2) log_b(b^x) = x, and (3) if x > 0, then b^(log_b(x)) = x. The paragraph uses these properties to solve examples of logarithmic expressions, such as log_10(10) = 1 and log_4(64) = 3. It also discusses the application of logarithms in real-life situations, like calculating the decibel level of a concert with a sound intensity of 10^-2 watts per square meter.

05:00

🔍 Applications of Logarithms in Chemistry

The second paragraph delves into the application of logarithms in chemistry, specifically in calculating the decibel level of a concert and the hydrogen ion concentration of vinegar with a pH level of 3.0. It uses the formula p = 10 * log(I/I_0) to find the decibel level, where I is the intensity of the sound and I_0 is a reference intensity. The result is a decibel level of 100 dB for the concert. For the vinegar, the paragraph uses the pH formula pH = -log[H+] to determine the hydrogen ion concentration, which is 10^-3.0 moles per liter. The paragraph concludes with a prompt for viewers to apply their knowledge of logarithms to solve five logarithmic expression problems, encouraging further learning and practice.

Keywords

💡Logarithms

Logarithms are mathematical operations that are the inverse of exponentiation, often used to express numbers in terms of their base and exponent. In the video, logarithms are the central theme, with a focus on their basic properties and how they can be applied to solve equations and real-life problems such as calculating decibel levels at concerts.

💡Base

In logarithms, the base is the number that is raised to the power of the exponent to get the result. It is crucial because it determines the scale of the logarithmic function. The video mentions that the base must be greater than zero and not equal to one, which are the conditions for a valid logarithm.

💡Properties of Logarithms

These are the fundamental rules that govern how logarithms behave. The video lists properties such as log_b(1) = 0 and log_b(b^x) = x, which are essential for simplifying and solving logarithmic expressions. Understanding these properties is key to the tutorial's educational purpose.

💡Logarithmic Equations

These are equations that involve logarithms. The video aims to teach viewers how to solve such equations by applying the basic properties of logarithms. An example given is finding the value of log expressions, which is a common task in mathematical and scientific applications.

💡Decibel Level

Decibels are a unit used to express the intensity of sound. The video provides an example of calculating the decibel level of a concert using logarithms, demonstrating how logarithmic properties can be applied to real-world scenarios to determine sound intensity.

💡pH Level

pH is a measure of how acidic or basic a solution is. The video explains how to calculate the hydrogen ion concentration of vinegar using the pH value and logarithms, showing the practical application of logarithms in chemistry.

💡Hydrogen Ion Concentration

This refers to the concentration of hydrogen ions (H+) in a solution, which is an important factor in determining the acidity or alkalinity of the solution. The video uses logarithms to calculate this concentration from a given pH value, illustrating the connection between mathematical concepts and chemical analysis.

💡Real-life Application

The video emphasizes the practical use of logarithms in everyday situations, such as calculating decibel levels at concerts and determining the acidity of vinegar. This demonstrates the relevance of mathematical concepts to real-world problems and their solutions.

💡Common Logarithm

A common logarithm is a logarithm with base 10. It is used in various scientific and engineering contexts, including the calculation of decibel levels as shown in the video. The common logarithm is a standard tool for expressing large or small numbers in a more manageable form.

💡Exponentiation

Exponentiation is the mathematical operation of raising a number to a power. It is the inverse operation of taking logarithms and is mentioned in the video as part of the explanation of logarithmic properties, such as log_b(b^x) = x.

💡Watts per Square Meter

This is a unit of measurement for power per unit area, often used in physics and engineering to describe the intensity of a force or energy distribution. The video uses this unit in the context of sound intensity at a concert, showing how logarithms can be applied to convert between different units of measurement.

Highlights

Objectives: Apply basic properties of logarithms and solve problems involving logarithmic equations.

Logarithm of one with any base b > 0 and b ≠ 1 is equal to zero.

Logarithm of b^x with base b is equal to x for x > 0.

b^(log_b(x)) = x for x > 0.

Using property one to find the value of log base 10 of 10.

Using property two to solve log base 4 of 64.

Logarithm of one is always zero, regardless of the base.

Calculating decibel levels using logarithms in a real-life situation.

Decibel level calculation formula: 10 * log(I/I0) where I0 is the reference intensity.

Example calculation: Decibel level of a concert with a sound intensity of 10^-2 watts/m^2.

Result of the decibel level calculation: 100 decibels.

Calculating hydrogen ion concentration using pH level.

pH level formula: pH = -log[H+] where [H+] is the hydrogen ion concentration.

Example calculation: Hydrogen ion concentration of vinegar with a pH of 3.0.

Result of the hydrogen ion concentration calculation: 10^-3.0 moles/liter.

Five practice questions provided to apply the basic properties of logarithms.

Encouragement to like, subscribe, and hit the bell button for more video tutorials.

Transcripts

[Music]

of

logarithms so our objectives

apply basic properties of logarithms and

solve

problems involving logarithmic equations

let b and x be real numbers such that b

is greater than zero and base net equal

to one

the basic properties of logarithm are as

follows

first your logarithm of one with base

b is equal to zero so tata and kapag one

and value nito

automatic in logarithmic n is equal to

zero

next logarithm of b raised to x

with base b is equal to x

exponent that is our logarithm so

unanswered goodnight n

if x is greater than zero then

b raised to the logarithm of x with base

b

so capacity okay in base net indito

and then in base theta that is basta

young x nothing greater than 0

the answer is

so that is equal to x okay

so 18 properties

so use the basic properties of

logarithms to find the value of the

following logarithmic expression

for example logarithms

so therefore so under property number

so unum property top property number two

so

one exponent and that is our logarithm

so therefore logarithm of ten

is equal to one under the property

number two now under property two

so ibiza b and uh same involuntary

base net and exponent nothing that is

our logarithm

and so that is under property two

next logarithm of 64 with base four

so panda nothing gagawin so c64 pretty

nothing expressed as

four cube bucket so but we multiply

nothing in that long base session four

four times four that is sixteen times

four sixty four

so same mean value nothing base net and

therefore

uh quantity is our exponentially

so log 64 with base four is equal to

three under property two okay

letter c five logarithm of two with

five raised to logarithm of two with

base five

so kitan kitana man so this

is our but if this is greater than zero

so therefore

uh and then antenating on your detox

number three okay next

logarithm of one so logarithm of one

so common logarithm but common

logarithms on base net and a ten

pero since one volume

under property number one

so

nito automatic last year mi

zero okay

i'll give you some uh problems noped

nothing

applying basic properties okay tomorrow

problem no

in a discussion that is a previous video

about application of logarithms in real

life situation

suppose you have seats to a concert

featuring your favorite musical artist

calculate the approximate decibel level

associated if a typical concert

sound intensity is 10 raised to negative

2

watts per square meter okay using the

formula

so uh

this is equal to 10 times the logarithm

of i

over 10 raised to negative 12.

so substitute nothing you'll give a nut

and a 10 raised to negative 2 density

not n

it will become 10 times logarithm of 10

raised to negative 2 so

papadi not internals a given attend over

10 raised to negative 12

and after that so since i mean based not

in and on gagoin's exponent capacity

divided

okay subtract nothing you exponent that

and so

negative 2 minus negative 12

so i'm getting on the end 10 raised to

10 okay say negative 2 minus negative 12

so i'm again negative plus 12 neon and

that is

positive 10 okay so

so under property number two so sub is a

property number two capac

since this is common logarithm

in detail that is our logarithm or good

nut in jan

so therefore that is 10 times 10

so 10 times 10 that is 100 a big sub

hand at concerts decibel level is 100

decibels okay

next calculate the hydrogen ion

concentration of vinegar has ph

level of 3.0 so it discussed then

uh doing some previous videos and about

application of logarithms so using this

formula

so ph is equal to so it on page level

we're going to identify

acidic neutral or basic

okay so negative logarithm so it

i i i don't know what symbol is this

booster in chemistry that is an ion

okay and after that we can substitute

the

event so in page level though nothing is

3.0

is equal to negative log and this

is

we need to multiply both sides by

negative one so making

negative three point zero so paramagne

positive

so that will become ten raised to

negative three point zero

is equal to ten raised to it

okay so hydrogen ion is equal to

10 raised to negative 3.0 therefore

the hydrogen ion concentration is 10

raised to negative 3.0 moles

per liter okay

okay using the basic properties of like

logarithms

find the value of the following

logarithmic expression so i'll give you

five questions here

and

okay so it is a good five questions so

perfect now congratulations in advance

betting

thank you for watching this video i hope

you learned something

don't forget to like subscribe and hit

the bell button

put updated ko for more video tutorial

this is your guide in learning your mod

lesson your walmart channel