Option C is correct,4.5 is greater than 3.8, so this solution does work.
If x represents the number of hours and y represents the number of liters of water, then we can plug the possible solutions into our inequality to see which solution(s) work.
Inequality
In mathematics, an inequality is a relation that makes a non-equal comparison between two numbers or other mathematical expressions. It is used most often to compare two numbers on the number line by their size.
The first option is having 0.5 liters of water for 4.5 hours of hiking. So will plug 3 in for y and 4.5 in for x:
y > 0.5x + 1.8
3 > 0.5(4.5) + 1.8
3 > 4.05
But 3 is not greater than 4.05, this solution does not work.
The next option is having 2 liters of water for 2.5 hrs of hiking:
2 > 0.5(2.5) + 1.8
2 > 3.05
But 2 is not greater than 3.05, so this solution does not work.
Option c is having 4.5 liters of water for 4 hours of hiking:
4.5 > 0.5(4) + 1.8
4.5 > 3.8
So 4.5 is greater than 3.8, this solution does work.
The last option is having 2.5 liters of water for 3 hours of hiking:
2.5 > 0.5(3) + 1.8
2.5 > 3.3
2.5 is not greater than 3.3, so this solution does not works
Option c is correct,4.5 is greater than 3.8 so this solution does work.
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y > 0.5x + 1.8
3 > 0.5(4.5) + 1.8
3 > 4.05
But 3 is not greater than 4.05, this solution does not work.
2 > 0.5(2.5) + 1.8
2 > 3.05
But 2 is not greater than 3.05, so this solution does not work.
4.5 > 0.5(4) + 1.8
4.5 > 3.8
2.5 > 0.5(3) + 1.8
2.5 > 3.3
2.5 is not greater than 3.3, so this solution does not works
Option c is correct,4.5 is greater than 3.8 so this solution does work.
A newsletter publisher believes that 43% of their readers own a Rolls Royce. A testing firm believes this is inaccurate and performs a test to dispute the publisher's claim. After performing a test at the 0.02 level of significance, the testing firm fails to reject the null hypothesis.
The conclusion regarding the test of the hypothesis is given as follows:
There is not sufficient evidence at the 0.02 level of significance that the percentage is not of 43%.What are the hypothesis tested?At the null hypothesis, it is tested if there is not enough evidence to conclude that the percentage is not of 43% = proportion of 0.43, hence:
[tex]H_0: p = 0.43[/tex]
At the alternative hypothesis, it is tested if there is enough evidence to conclude that the percentage is not of 43%, hence:
[tex]H_1: p \neq 0.43[/tex]
The decision was to not reject the null hypothesis, meaning that there is not sufficient evidence at the 0.02 level of significance that the percentage is not of 43%.
Missing InformationThe problem asks to identify the correct conclusion from these two options:
There is not sufficient evidence at the 0.02 level of significance that the percentage is not of 43%.There is sufficient evidence at the 0.02 level of significance that the percentage is not of 43%.More can be learned about the test of an hypothesis at https://brainly.com/question/13873630
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Colette bought 1/4 of a pound of green grapes and 1/8 of a pound of red grapes. How much more did the green grapes weigh than the red grapes?
The green grapes weigh 0.125 pounds more than the red grapes.
Step-by-step explanation:1/4 of a pound = 0.25 pounds
1/8 of a pound = 0.125 pounds
0.25 - 0.125 = 0.125 pounds
Hope this helps :)
Keyshawn is asked to compare and contrast the domain and range of two functions F(x)=5x g(x)=5^x which statements could he include in his explanation
Domain is the input and range is the output of a function. By considering the options The domain of both function is all real number and The range of g(x) >0 are the correct answers.
What is Domain?Domain is the input of a function.
What is range?Range is the output of a function.
The domain of both functions is the set of all real numbers, since both functions are defined for all values of x.
The range of the function g(x)=5^x is the set of all positive real numbers, since any positive real number can be obtained by raising 5 to the power of a real number
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What type of number is -560,114?
Choose all answers that apply:
A Whole number
B
C
D
Integer
Rational
Irrational
Answer:
It is an integer.
It is rational.
Step-by-step explanation:
A whole number is in the set
0, 1, 2, 3...
This set starts at zero and only includes zero and the positive numbers. So the given number is NOT a whole number.
It IS an integer. Integers include positive and negative numbers. That is, whole numbers and their opposites.
The number IS rational. That is, it can be written as a ratio:
-560114/1
Since it is rational, it is not irrational.
Suppose the COMBINED area is known to be 0.10416, assume there is equal area on each side. Determine the corresponding Z-scores.A) z=±1.62 B) z=±1.58 C) z=±1.72 D) z=±1.66 E) z=±1.69 F) z=±1.49 G) None of These
An combined area of 0.10416 to the right means that it must have an area of 0.90 to the left, so the answer is again 1.28.
Every normally distributed random variable has a slightly different distribution shape, the only way to find areas using a table is to standardize the variable - transform our variable so it has a mean of 0 and a standard deviation of 1. How do we do that? Use the z-score!
Z = ( x - μ)/σ
if the random variable X has a mean μ and standard deviation σ,
then transforming X using the z-score creates a random variable with mean 0 and standard deviation 1!
With that in mind, we just need to learn how to find areas under the standard normal curve, which can then be applied to any normally distributed random variable.
An area of 0.10416 to the right means that it must have an area of 0.90 to the left, so the answer is again 1.28.
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Every normally distributed random variable has a slightly different distribution shape, the only way to find areas using a table is to standardize the variable - transform our variable so it has a mean of 0 and a standard deviation of 1. How do we do that? Use the z-score!
Z = ( x - μ)/σ
if the random variable X has a mean μ and standard deviation σ,
An area of 0.10416 to the right means that it must have an area of 0.90 to the left, so the answer is again 1.28.
A train travelling along a straight line with constant acceleration is observed to travel consecutive distances of 1km in times of 30s and 60s respectively. Find the initial velocity of the train.
Answer:
116.7m/s
Step-by-step explanation:
distance(s) = 1km = 1 × 1000m = 1000m
Time(t) = 60– 30 = 30s
initial velocity = u
acceleration (a) = 10m/s
using the formula
S = ut + ½at²
1000 = 30u + ½ × 10 × 30²
1000 = 30u + 5 × 900
1000 = 30u + 4500
30u = 4500 – 1000
30u = 3500
divide both sides by 30
u = 116.7m/s
i hope i helped
The table shows the colour of the bicycles in
a bike shed.
Bookwork code: S79 EX
<
< Back
Colour Frequency
Silver
3
Green
1
a) What fraction of the bicycles are silver?
Give your answer in its simplest form.
b) How many sections on this pie chart
would you shade to show the bicycles that
are silver?
✓ Scroll down
Watch video
Answer >
A fraction of the bicycles which are silver is 3/4.
You should shade three (3) sections on the pie chart to show the number of bicycles that are silver.
What is a pie chart?In Mathematics, a pie chart can be defined as a type of graph that is used for the graphical representation of the proportion relationship between each part or unit to a whole, especially by dividing a circle into various sectors.
Based on the information provided, we can logically deduce the following parameters;
Number of silver bicycles = 3 silver bicycles.
Number of green bicycles = 1 green bicycles.
Therefore, the total number of bicycles is equal to 4 bicycles.
For the fraction of silver bicycles, we have;
Fraction = 3/4 bicycles.
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What is the y/x ratio
The angles whose measure are greater than the measure of ∠1 are ∠2, ∠3, ∠7, ∠5 .
What is the general equation of a straight line?The general equation of a straight line is -
y = mx + cwhere -
[m] : slope[c] : y - interceptGiven is a graph as shown in the image.
The ratio of {y/x} is called the slope of a line. So, we can write -
m = 3/2
m = 1.5
Therefore, the ratio of [y] by [x] is equivalent to 1.5.
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There were 616 tickets purchased for a major league baseball game. The lower box tickets cost $12.50 and the upper reserved tickets cost $8.00. The total amount of money spent was $6543.50. How many of each kind of ticket were purchased?
The number of the tickets purchased for upper reserve is 146 and lower is 470.
What is an expression?
The mathematical expression combines numerical variables and operations denoted by addition, subtraction, multiplication, and division signs.
Mathematical symbols can be used to represent numbers (constants), variables, operations, functions, brackets, punctuation, and grouping. They can also denote the logical syntax's operation order and other properties.
Given that there were 616 tickets purchased for a major league baseball game. The lower box tickets cost $12.50 and the upper reserved tickets cost $8.00. The total amount of money spent was $6543.50.
The equations will be formed as,
L + U = 616
12.5L + 8U = 6543.5
Solve the above equations,
12.50L + 8 (616-L) = 6543.5
12.5L + 4928 - 8L = 6543.5
12.50L - 8L = 6543.5 - 4428
4.50L = 2118.5
L = 470
U = 616 - 470
U = 146
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Miguel earns extra money working two weekends with his dad. He is saving to buy a new bike that costs $140. Heather says that Miguel needs to earn more than $6 for each hour that he works to have enough money to buy the bike. Her work is shown below. Write an inequality to explain why she is incorrect
An inequality to explain the reason Heather is incorrect is t > $8.75.
What is an inequality?In Mathematics, an inequality can be defined as a mathematical relation that is typically used for comparing two (2) or more numerical data and variables in an algebraic equation based on any of the following inequality symbols:
Less than (<).Greater than (>).Greater than or equal to (≥).Less than or equal to (≤).From the histogram, we can logically deduce the following information;
Miguel’s work record during Weekend 1 = 9 hours.
Miguel’s work record during Weekend 2 = 7 hours.
Therefore, the total number of hours which Miguel worked in two Weekends is given by;
Total hours = 9 hours + 7 hours
Total hours = 16 hours.
Next, we would calculate the amount of money that Miguel would earn per hour;
Amount = $140/16
Amount = $8.75 per hour.
Note: Let the variable t represent the total earnings.
In conclusion, the correct inequality is t > $8.75.
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Create a multiplication equation with a solution of 2/7
Answer:
1 / 7 · 2 = 2 / 7
The initial state of a queuing system (for example, when a restaurant first opens), is referred to as the ____ state. primary steady start-up transientintroductory
The initial state of a queuing system (for example, when a restaurant first opens), is referred to as the primary state.
In the queue system, what are the three states?1) Orderly queue, also known as FIFO (First In, First Out) or FCFS (First Come, First Serve).
2) Stack, also known as LCFS (Last Come First Serve), or LIFO (Last In First Out).
3) SIRO (Serve in Irregular Request).
What is the queuing system's steady state?Let P n(t) represent the probability of having n customers in the system at time t. This is the steady state of a queueing system, where the probability of the number of customers in the system is independent of t.
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Harry and Tim both made New Year’s Resolution. Harry made 5 more resolutions than Tim. Together they made 13 resolutions. How many resolutions did Harry make?
Answer:
Harry made 9 resolutions.
Step-by-step explanation:
Tim made x resolutions.
Harry made five more, x + 5.
Harry and Tim's together was 13.
x + x + 5 = 13
combine like terms.
2x + 5 = 13
subtract 5
2x = 8
divide by 2
x = 4
Tim made 4 resolutions.
Harry made 4 + 5, that is, 9 resolutions.
Check:
Harry and Tim together is 13.
4 + 9 = 13
Harry made 9 resolutions.
A service club agrees to donate at least 500 hours of community service. A full member is to give 4 hours, and a pledge gives 6 hours.
(a) Write the inequality that expresses this information.
(b) Graph the inequality.
(a) 4x + 6y ≥ 500 , (b) The graph of this inequality is a line with a slope of 4/6 and a y-intercept of 500/6.
(a) The inequality expresses the total hours of community service that must be given, which is at least 500 hours. The 4x term represents the hours given by the full members, and the 6y term represents the hours given by the pledges. The ≥ symbol means that the sum of these two terms must be greater than or equal to 500.
(b) To graph the inequality, you first need to find the y-intercept, which is 500/6. Then you need to find the slope, which is 4/6. This means that the line has a positive slope and passes through the point (0, 500/6). The graph of the inequality is a line that passes through this point and has a positive slope of 4/6.
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Which of the following expressions has the leftmost value on the number line?
A.-7-10 O
B. 7-10
C.10-(-7)
D.10-7
Answer:
A.-7-100. [Since it is -107]
May someone help me with this.
Step-by-step explanation:
since I don't see the options, I can only tell you what it is supposed to be and you need to pick the best fitting option :
the score of the test is simply
the number of correctly answered questions times 3 (as every correctly answered question is with 3 points).
the equation is therefore
y = R(x) = 3x
the independent variable x represents the number of correctly answered questions,
and the dependent variable is y, the test score, because the test score depends on the number of correctly answered questions.
so,
R(25) = 3×25 = 75, meaning 25 correctly answered questions × 3 results in the score of 75.
What is the sum of the interior angles of a regular polygon with 9 sides?
1,260°
1,620°
1,800°
1,980°
The sum of the interior angles of a regular polygon with 9 sides will be 1260°. The correct option is A.
What is the angle of the polygons?An angle is defined in mathematics as the figure formed by joining two rays at their common endpoint. An interior angle is an angle that exists within a shape. Polygons are closed shapes with sides and vertices. All of the interior angles of a regular polygon are equal.
Given that the number of the sides of the polygon is 9. The sum of the interior angles of the polygons will be calculated as,
Sum = ( n - 2 ) 180°
Sum = ( 9 - 2 ) 180
Sum = 7 x 180
Sum = 1260°
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A line passes through the point (4, -8) and has a slope of 5/2
Write an equation in slope-intercept form for this line.
help me pls this is due in 30 minutes
y=5/2x+24
Step-by-step explanation:
use the formula y-y1=m(x-x1)
5/2 symbolizes m
put in 4 for y1 and put in -8 for x1
y-(4)=(5/2)(x-(-8)
y-4=5/2(x+8)
y-4=5/2x+20
y=5/2x+24
Answer:
y = 5/2x -18
Step-by-step explanation:
You want the slope-intercept equation for the line with slope 5/2 through point (4, -8).
Point-slope formIt can work well to start with the point-slope equation and rearrange it to the desired form.
y -k = m(x -h) . . . . . . . line with slope m through point (h, k)
y -(-8) = 5/2(x -4) . . . . . line with slope 5/2 through point (4, -8)
Slope-intercept formAdd -8 and eliminate parentheses
y = 5/2x -10 -8
y = 5/2x -18 . . . . . . . . simplify
The desired equation is y = 5/2x -18.
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Let Q represent a mass (in grams) of carbon (^14C), whose half-life is 5,715 years. The quantity of carbon-14 present after t years is given by the following
Q = 11(1/2)^(t/5715)
a. Determine the initial quantity (when t=0) g
b. Determine the quantity present after 2000 years. (Round your answer to 2 decimal places).
c. What is the graph of the function over the interval t = 0 to t = 10,000
Answer:
(a) 11 g
(b) 8.63 g
(c) Attached
Step-by-step explanation:
Given function:
[tex]Q = 11 \left(\dfrac{1}{2}\right)^{\dfrac{t}{5715}}[/tex]
where:
Q = quantity of carbon-14 presentt = time (in years)Part (a)Substitute t = 0 into the given function to determine the initial quantity:
[tex]\begin{aligned} t=0 \implies Q &= 11 \left(\dfrac{1}{2}\right)^{\dfrac{0}{5715}}\\\\&=11\left(\dfrac{1}{2}\right)^0\\\\&=11\left(1\right)\\\\&=11\; \rm g\end{aligned}[/tex]
Part (b)Substitute t = 2000 into the given function to determine the quantity present after 2000 years:
[tex]\begin{aligned} t=2000 \implies Q &= 11 \left(\dfrac{1}{2}\right)^{\dfrac{2000}{5715}}\\\\&=11\left(0.784697887...\right)\\\\&=8.63\; \rm g\;\; \sf (2 \; d.p.)\end{aligned}[/tex]
Part (c)[tex]\begin{aligned} t=10000 \implies Q &= 11 \left(\dfrac{1}{2}\right)^{\dfrac{10000}{5715}}\\\\&=11\left(0.297346855...\right)\\\\&=3.27\; \rm g\;\; \sf (2 \; d.p.)\end{aligned}[/tex]
The graph of the function over the interval t = 0 to t = 10000 is attached.
The terminal side of an angle θ in standard position intersects the unit circle at (√30/7,-√19/7). What is cot(θ)?
The cotangent of the angle is -√570/30
How to determine the cotangent of the angle?From the question, we have the following parameters that can be used in our computation:
(√30/7,-√19/7)
This means that
(x, y) = (√30/7,-√19/7)
The cot(θ) is calculated as
cot(θ) = y/x
Substitute the known values in the above equation, so, we have the following representation
cot(θ) = (-√19/7)/(√30/7)
Evaluate
cot(θ) = -√19/√30
Rationalize
cot(θ) = -√570/30
Hence, the value of cot(θ) is -√570/30
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Question 1 of 9
Use the scale to help you solve the equation and find the value of x. Enter the
value of x below. x+8=10
Answer: x + 8 - 8 = 10 - 8
Simplifying the left side of the equation gives us:
x = 2
Therefore, the value of x in this equation is 2.
At Jalisco Electronics, assemblers are paid according to the following differential piece 4. rate scale: 1–20 units in a day, $4.50 each; 21–30 units, $5.50 each; and $7 each for every
unit over 30. Adrian Ortega assembled 70 units in one week. Find his gross pay.
The gross pay of Adrian Ortega for assembling 70 units of electronics in a week is $425
What is his gross pay?Gross pay refers to the total payment earned over a period of time.
1–20 units in a day = $4.50 each21–30 units = $5.50 eachOver 30 units = $7 eachAdrian Ortega assembled 70 units in one week
Pay for 1–20 units = $4.50 × 20
= $90
Pay for 21 - 30 units = $5.50 × 10
= $55
Pay for 30 - 70 = $7 × 40
= $280
Gross pay = $90 + $55+ $280
= $425
Therefore, Adrian Ortega is paid $425 in a week for assembling 70 electronics.
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What is the equation in standard form of the line that passes through the
point (4, −8) and 1 has a slope of 1/4?
The equation in standard form of the line that has a slope of 1/4 and passes through the point (4, −8) is written as: -x + 4y = -36.
What is the Equation of a Line in Standard Form?The equation of a line can be written in standard form as Ax + By = C.
Given the following:
A point (4, -8) = (a, b).
Slope (m) = 1/4.
Substitute m = 1/4 and (a, b) = (4, -8) into y - b = m(x - a):
y - (-8) = 1/4(x - 4)
y + 8 = 1/4(x - 4)
Rewrite the equation in standard form:
4(y + 8) = x - 4
4y + 32 = x - 4
4y = x - 4 - 32
4y = x - 36
-x + 4y = -36
The equation is: -x + 4y = -36.
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Which equation, when solved, results in a different value of x than the other three?
Equation of option (d) is having different value of x than other 3 equations.
So, option (d) is correct option.
How to solve equation?Write down the problem.Remember the order of operations: DMAS, which stands for Multiplication/Division, and Addition/SubtractionDo the multiplication.Do the addition and subtraction.Keep the variable at L.H.S and alone.To do this, just divide both sides of the equation by coefficient of x to find x.Solving equations
(a) (-7/8)x-(3/4)=20
adding 3/4 on both sides, we get
(-7/8)x = 20+(3/4)
multiplying (-8/7) on both sides, we get
x = -23.71
(b) (7/8)x+(3/4)=-20
subtracting 3/4 on both sides, we get
(7/8)x = -20-(3/4)
multiplying (8/7) on both sides, we get
x = -23.71
(c) 7(-1/8)x-(3/4)=20
adding 3/4 on both sides, we get
(-7/8)x = +20+(3/4)
multiplying (-8/7) on both sides, we get
x = -23.71
(d) (-7/8)×(-8/7)x-(3/4)=20×(-8/7)
⇒ x-(3/4) = -22.85
adding 3/4 on both sides, we get
x = -22.85+(3/4)
x = -22.10
So equation in option (d) has different value of x than all other equation.
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Find the point of intersection between a line that is tangent to the function
() = x^2 + 4 at = −1 ,
and the function
() = x^2 − 4
There is no point of intersection between a line that is tangent to the function () = x² + 4 at = −1 , and the function () = x² − 4
What is the point of intersection about?To find the point of intersection between the line that is tangent to the function y = x² + 4 at x = −1 and the function y = x² − 4 , we can set the two equations equal to each other and solve for x. This will give us the x-coordinate of the point of intersection.
Setting the two equations equal to each other, we have:
x^2 + 4 = x² - 4
This simplifies to:
2 = -8
This equation has no solutions, so the two functions do not intersect. This means that the point of intersection does not exist.
Alternatively, we could see that the two functions have different slopes at the point x = −1 .
The function y = x² + 4 has a slope of 2x at this point, which is equal to -2 . The function y = x² − 4 has a slope of 2x , which is equal to -2 at this point as well.However, since the two functions have different y-intercepts, they cannot be tangent to each other at this point. Therefore, the point of intersection does not exist.
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please help with this
The equation variation for the electric circuit is V = 25R.
How to find the equation variation of an electric circuit?In each electric circuit, the voltage in a battery is measured in volt and the resistance in the circuit is measured in ohms.
The voltage, V is directly proportional to the resistance, R. In a given circuit the voltage is 200 volts, when the resistance is 8 ohms.
The equation variation for the circuit can be represented as follows:
V α R
V = kR
where
k = constant of proportionalityTherefore,
voltage = 200 volts
resistance = 8 ohms
200 = 8k
k = 200 / 8
k = 25
Therefore, the equation is V = 25R
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Determine which of the lines, if any, are parallel or perpendicular. Explain.
6. Line a passes through (0, 4) and (4, 3).
Line b passes through (0, 1) and (4, 0).
Line c passes through (2, 0) and (4, 4).
Answer:
Step-by-step explanation:
c 0,4 .01
Q30: Work out the surface of the following prism
Surface area of the given prism is 6 square cm .
How to Calculate the Surface Area of Prism?
The steps to determine the surface area of the prism are:
Step 1: Note down the given dimensions of the prism.
Step 2: Substitute the dimensions in the surface area of prism formula (2 × Base Area) + (Base perimeter × height).
Step 3: The value of the surface area of the prism is obtained and the unit of the surface area of the prism is placed in the end (in terms of square units).
The lateral area is the area of the vertical faces, in case a prism has its bases facing up and down. Thus, the lateral surface area of prism = base perimeter × height
The total surface area of a Prism = Lateral surface area of prism + area of the two bases = (2 × Base Area) + Lateral surface area or (2 × Base Area) + (Base perimeter × height).
Area of triangular cross-section:
Area = 1/2 . b . h
Area = 1/2 . 4. 3
Area = 6 square cm
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Which expression is equivalent to 8(6s+6)?
Answer:
48s + 48
Step-by-step explanation:
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Rewrite the logarithm as a ratio of common logarithms and natural logarithms.
log_1/3(4)
a. common logarithms
b. natural logarithms
Common logarithm is the logarithm with the base 10, it is written as:
log a, lg a or log₁₀ aNatural logarithm is the logarithm with the base of e and is written as ln a.
How to rewrite a logarithm [tex]log_a\ b[/tex] as a ratio of logarithms with a different base c:
[tex]log_a\ b=log_c\ b\ /\ log_c\ a[/tex]Apply this to the given logarithm:
a. Common logarithms:
[tex]log_{1/3}\ 4=log\ 4\ /\ log\ (1/3)[/tex]b. Natural logarithms:
[tex]log_{1/3}\ 4=ln\ 4\ /\ ln\ (1/3)[/tex]Answer:
[tex]\textsf{a)} \quad \dfrac{\log_{10}4}{\log_{10}\dfrac{1}{3}}=-\dfrac{\log_{10}4}{\log_{10}3}[/tex]
[tex]\textsf{b)} \quad \dfrac{\ln 4}{\ln \dfrac{1}{3}}=-\dfrac{\ln 4}{\ln 3}[/tex]
Step-by-step explanation:
Given logarithm:
[tex]\log_{\frac{1}{3}}(4)[/tex]
Part (a)The common logarithm is the logarithm with base 10.
[tex]\boxed{\textsf{Change of base}: \quad \log_ba=\dfrac{\log_xa}{\log_xb}}[/tex]
Use the change of base formula to rewrite the given logarithm as a ratio of common logarithms:
[tex]\implies \log_{\frac{1}{3}}(4)=\dfrac{\log_{10}4}{\log_{10}\dfrac{1}{3}}[/tex]
[tex]\boxed{\begin{minipage}{5cm} \underline{Log quotient law}\\\\$\log_a \left(\dfrac{x}{y}\right)=\log_ax - \log_ay$\\\end{minipage}}[/tex]
This can be simplified using the log quotient rule:
[tex]\implies \log_{\frac{1}{3}}(4)=\dfrac{\log_{10}4}{\log_{10}\dfrac{1}{3}}[/tex]
[tex]\implies \log_{\frac{1}{3}}(4)=\dfrac{\log_{10}4}{\log_{10}1-\log_{10}3}[/tex]
[tex]\implies \log_{\frac{1}{3}}(4)=\dfrac{\log_{10}4}{0-\log_{10}3}[/tex]
[tex]\implies \log_{\frac{1}{3}}(4)=-\dfrac{\log_{10}4}{\log_{10}3}[/tex]
Part (b)The natural logarithm is the logarithm with base e.
[tex]\textsf{Also} \; \log_ex=\ln x[/tex]
Use the change of base formula to rewrite the given logarithm as a ratio of natural logarithms:
[tex]\implies \log_{\frac{1}{3}}(4)=\dfrac{\log_{e}4}{\log_{e}\dfrac{1}{3}}[/tex]
[tex]\implies \log_{\frac{1}{3}}(4)=\dfrac{\ln 4}{\ln \dfrac{1}{3}}[/tex]
[tex]\boxed{\begin{minipage}{5cm} \underline{Log quotient law}\\\\$\ln \left(\dfrac{x}{y}\right)=\ln x - \ln y$\\\end{minipage}}[/tex]
This can be simplified using the log quotient rule:
[tex]\implies \log_{\frac{1}{3}}(4)=\dfrac{\ln 4}{\ln \dfrac{1}{3}}[/tex]
[tex]\implies \log_{\frac{1}{3}}(4)=\dfrac{\ln 4}{\ln1 - \ln 3}[/tex]
[tex]\implies \log_{\frac{1}{3}}(4)=\dfrac{\ln 4}{0 - \ln 3}[/tex]
[tex]\implies \log_{\frac{1}{3}}(4)=-\dfrac{\ln 4}{\ln 3}[/tex]