BASIC PROPERTIES OF LOGARITHMS || GRADE 11 GENERAL MATHEMATICS Q1
Summary
TLDRThis 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
- π The video focuses on the application of basic properties of logarithms to solve logarithmic equations.
- π’ Logarithm of one with any base \( b > 0 \) and \( b \neq 1 \) is zero, expressed as \( \log_b(1) = 0 \).
- π The logarithm of \( b^x \) with base \( b \) is equal to \( x \), or \( \log_b(b^x) = x \).
- β¬οΈ If \( x > 0 \), then \( b \) raised to the logarithm of \( x \) with base \( b \) equals \( x \), or \( b^{\log_b(x)} = x \).
- π The video provides examples to illustrate the properties, such as calculating \( \log_{10}(10) = 1 \) and \( \log_4(64) = 3 \).
- πΆ 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.
- π― 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.
- π The presenter encourages viewers to apply these basic properties to solve five logarithmic expression problems presented at the end of the video.
- π The video concludes with a prompt for viewers to like, subscribe, and hit the bell button for more educational content.
- π 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 β 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
π 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.
π 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.
Mindmap
Keywords
π‘Logarithms
π‘Base
π‘Properties of Logarithms
π‘Logarithmic Equations
π‘Decibel Level
π‘pH Level
π‘Hydrogen Ion Concentration
π‘Real-life Application
π‘Common Logarithm
π‘Exponentiation
π‘Watts per Square Meter
Highlights
Objectives: Apply basic properties of logarithms and solve problems involving logarithmic equations.
Logarithm of one with any base b > 0 and b β 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
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