The logarithmic graph for the function loga(x) represents relationships similar to that of a log-log plot such as exponential relationships. Understanding properties of logarithms and principles of graphing can help interpret this function. Unlike linear equations, these graphs have a unique shape that are not straight lines.
Explanation:The graph of the logarithmic function f(x) = loga(x) can be understood using logarithmic properties and graph principles. The log-log plot reveals that logarithmic graphs represent exponential relationships such as the decay of radioactivity or the relationship between light's frequency and wavelength. The logarithmic scale is advantageous in graphing large ranges of values.
There are a few principles to note about logarithms for this. For instance, the logarithm of a number with an exponent is equal to the exponent multiplied by the logarithm of the number. Similarly, the logarithm resulting from dividing two numbers is the difference between the logarithms of the two numbers. These properties can be helpful in interpreting the logarithmic graph.
Our graph, however, is not a straight line as in the case of linear equations which have a uniform slope, a y-intercept and x as the horizontal axis. Logarithmic graphs have unique characteristics determined by the base and the values of x.
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30 for 48.30 or 32 for 49.60
Conplete the square to solve the quadratic equation. x
[tex]1 {x}^{2} - 6x + 12 = 2x + 20[/tex]
Answer:
[tex]\large\boxed{x=4\pm2\sqrt6}[/tex]
Step-by-step explanation:
[tex]x^2-6x+12=2x+20\qquad\text{subtract}\ 2x\ \text{and}\ 12\ \text{from both sides}\\\\x^2-8x=8\\\\x^2-2(x)(4)=8\qquad\text{add}\ 4^2=16\ \text{to both sides}\\\\x^2-2(x)(4)+4^2=8+4^2\qquad\text{use}\ (a-b)^2=a^2-2ab+b^2\\\\(x-4)^2=24\iff x-4=\pm\sqrt{24}\\\\x-4=\pm\sqrt{4\cdot6}\\\\x-4=\pm\sqrt4\cdot\sqrt6\\\\x-4=\pm2\sqrt6\qquad\text{add 4 to both sides}\\\\x=4\pm2\sqrt6[/tex]
Final answer:
The quadratic equation x² - 6x + 12 = 2x + 20 is solved by completing the square, resulting in the solution x = 4 ± √24.
Explanation:
To solve the quadratic equation x² - 6x + 12 = 2x + 20 by completing the square, we first need to rearrange the equation to get 0 on one side. We do this by subtracting 2x and 20 from both sides, which gives us:
x² - 8x - 8 = 0
Next, we find the value to complete the square for the x terms. We take half of the coefficient of x, which is 8/2 = 4, and then square it, getting 4² = 16. We add 16 to both sides of the equation to form a perfect square on the left:
x² - 8x + 16 = 8 + 16
This simplifies to:
(x - 4)² = 24
To find the value of x, we take the square root of both sides:
x - 4 = ±√24
Finally, we solve for x:
x = 4 ± √24
The figures shown are similar. What is the scale factor? Diagram contains two trapezoids. The first has a base of 21. The second has a base of 14.
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In a random sample of 160 high school students, 64% of the students participated in summer camp activities. The estimate had a margin of error of 2.5% at a 95% confidence level. Which of the following is a potential percentage of high school students who participated in summer camp activities.
A. 60%
B. 68%
C. 62%
D. 67%
Please answer 2 3 4 and explain how you got your answer I will give brainliest must explain for all 3
Which polynomial does this sum of tiles represent?A
Answer:
The polynomial result sum is [tex]2x^{2} +4x+6[/tex]
Step-by-step explanation:
To calculate the sum of the polynomial expressions first we have to write individually each one and sum the terms of it:
Fisrt one: [tex]x^{2} + x + x+ 1 + 1 = x^{2} +2x+2[/tex]
Second one: [tex]x^{2} +x+x+1+1+1+1=x^{2} +2x+4[/tex]
Now having both equations in simplified form we can sum them:
[tex]x^{2} +2x+2+x^{2} +2x+4=2x^{2} +4x+6[/tex]
So the sum of the polynomial equations is: [tex]2x^{2} +4x+6[/tex]
Help with 8!!!!! Plz
The lines shown below are parallel. If the green line has a slope of -3/7, what is the slope of the red line?
For the given condition of lines to be parallel, slope of red line is same as green line which is equal to [tex]\frac{-3}{7}[/tex].
What are parallel lines?" Parallel lines are defined as the lines in the same plane with equal distance between them and do not intersect in the same plane."
What is the slope?" Slope is defined as the ratio of increment in the y-coordinates to the x-coordinates."
Condition of slope for two parallel lines
[tex]y = m_{1} x + c_{1}\\\\y = m_{2} x + c_{2}[/tex]
[tex]m_{1}[/tex] and [tex]m_{2}[/tex] represents the slope of parallel lines.
As slope of parallel lines always equal.
Therefore,
[tex]m_{1} = m_{2}[/tex]
According to the question,
Given,
The slope of the green line [tex]= \frac{-3}{7}[/tex]
Condition given,
Lines are parallel.
Using the condition of slope of parallel lines we have,
The slope of the parallel lines is always equal.
Therefore,
The slope of the red line [tex]= \frac{-3}{7}[/tex]
Hence, the slope of the red line [tex]= \frac{-3}{7}[/tex] as per the given lines is parallel.
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The regular price of a Space Invader game is $52, but it’s on sale. The discount is $13. What percent discount is the Space Invader game?
This graph shows the linear relationship between the time in seconds, x, for Caroline's walk and the distance in meters, y, Caroline is from her starting point. The graph passes through the point (20,30). According to the graph, how many meters will Caroline walk in exactly one hour?
Answer:
5400
Step-by-step explanation:
There are 3600 seconds in a hour.
y=1.5(3600)
y=5400
Find the supplement of an angle that measures 20 degrees.
A) 50 degrees
B) 70 degrees
C) 90 degrees
D) 160 degrees
The supplement of an angle that measures 20 degrees will be 160°. Then the correct option is D.
What is an angle?The inclination is the separation seen between planes or vectors that meet. Degrees are another way to indicate the slope. For a full rotation, the angle is 360°.
Supplementary angle - Two angles are said to be supplementary angles if their sum is 180 degrees.
The one angle is 20 degrees. Then the other angle is given as,
x + 20° = 180°
x = 180° - 20°
x = 160°
The supplement of an angle that measures 20 degrees will be 160°. Then the correct option is D.
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A right rectangular prism has a length of 7 centimeters, a width of 5 centimeters, and a height of 4 centimeters. What is the volume of the prism? Enter your answer in the box.
|____|cm³
The volume of the right rectangular prism with dimensions of 7 cm by 5 cm by 4 cm is 140 cubic centimeters (140 cm³).
To calculate the volume of a right rectangular prism, you can use the formula: Volume (V) = length (l) × width (w) × height (h). For a prism with a length of 7 centimeters, a width of 5 centimeters, and a height of 4 centimeters, the volume is calculated as follows:
V = 7 cm × 5 cm × 4 cm
V = 35 cm × 4 cm
V = 140 cm³
So, the volume of the prism is 140 cubic centimeters (140 cm³).
Aldo must choose a number between 55 and 101 that is a multiple of 3, 6, and 7.
what measure of spread is LEAST by the long tail in the graph?
Answer:
IQR
Step-by-step explanation:
Interquartile range (IQR): it's the difference between the largest and smallest values in the middle 50% of a set of data. It is not affected by the long tail in the graph because its calculation only consider the central part of data.
Range: it’s the difference between the highest value and the lowest value in a data set. It is affected by long tails because consider the whole data
Median absolute deviation (MAD): it's the median of the absolute deviations from the data's median. It's moderate affected by long tails because atypical measures give greater absolute deviation that may cause a greater MAD.
What is the value of the expression |48÷(-6)|+|-35÷7|
write a expression showing the sum of 8 and a number f
is each expression equivalent to 2x^2 - 50?
a 2(x-5)(x+5)
b -2(x+5)(x+5)
c -2(5-x)(x+5)
Say yes or no for each one
Final answer:
The original expression 2x² - 50 is equivalent to option a) 2(x-5)(x+5) and option c) -2(5-x)(x+5), but not to option b) -2(x+5)(x+5).
Explanation:
The student has asked whether each given expression is equivalent to 2x² - 50.
a) 2(x-5)(x+5): Yes, this is equivalent to 2x² - 50 because when you apply the difference of squares formula, you get 2(x² - 25), which simplifies to 2x² - 50.
b) -2(x+5)(x+5): No, this results in -2(x² + 10x + 25), which simplifies to -2x² - 20x - 50, and is not equivalent to 2x² - 50.
c) -2(5-x)(x+5): No, this expression simplifies to -2(-x² - 5x + 5x + 25) or 2x² - 50, which is actually equivalent to the original expression. The negative sign inside the parentheses changes the sign of x when multiplied by the outside negative factor, leading to a positive result for the x² term.
A merchant buys a television for $125 and sells it for a retail price of $200. What is the markup
The markup is 60%.
Given that,
A merchant buys a television for $125 and sells it for a retail price of $200.
Based on the above information, the calculation is as follows:
[tex]= (\$200 - \$125)\div \$125[/tex]
= 60%
Therefore we can conclude that The markup is 60%.
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what is the area of a circumference 37.7
The area of circle of circumference 37.7 is 112.76 .
What is area of circle?The area of a circle is pi times the radius squared (A = π r²)
where ,
A = Area of circle
r = radius of circle
What is circumference of circle?The circumference of a circle is defined as the linear distance around it.
C=2πr
where ,
c = circumference of circle
r = radius of circle
According to question
Circumference = 37.7
2[tex]\pi r\\[/tex] = 37.7
[tex]\pi r[/tex] = 18.85
[tex]\frac{22}{7}r[/tex]=18.85 ([tex]\pi = 22/7[/tex])
r = 18.85 × [tex]\frac{7}{22}[/tex]
r = 5.99
Now area of circle = [tex]\pi r^{2}[/tex]
Area of circle = [tex]\frac{22}{7} * 5.99^{2}[/tex]
Area of circle = 112.76
Hence , the area of circle is 112.76 .
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A baseball team played 147 regular season games. The ratio of the number of games they won to the number of games they lost was four thirds 4/3. How many games did they win? How many games did they lose?
Final answer:
To find the number of games won and lost by the baseball team, use the given win to loss ratio of 4/3 and the total number of games played (147) to establish a system of equations and solve for the number of wins and losses. The team won 84 games and lost 63 games.
Explanation:
The student has asked how many games a baseball team won and lost given that they played 147 regular season games with a win to loss ratio of 4/3. To solve this, let w represent the number of games won and l represent the number of games lost. According to the ratio, we have w/l = 4/3. Because the total number of games is 147, we can express this as w + l = 147.
First, express the number of losses in terms of wins using the ratio: l = 3w/4.
Next, substitute l in the total games equation: w + 3w/4 = 147.
Solving for w gives w = 84 (number of games won).
To find l, substitute w back into l = 3w/4 to get l = 63 (number of games lost).
The baseball team won 84 games and lost 63 games.
Demostrar que si las magnitudes de la suma y la diferencia de dos vectores son iguales, entonces los vectores son perpendiculares.
Which of the following are factors of 6x^2+30x-36?
A) x-1
B)x+2
C)2
D)3
E)x+6
F)x-6
G)x-2
Enter the explicit rule for the geometric sequence. 120, 40, 403, 409, 4027, ...
a(n)=
The explicit rule for the given geometric sequence is [tex]a(n) = 120 \times (1/3)^{n-1}[/tex].
A geometric sequence is a sequence in which each term after the first is found by multiplying the previous term by a constant called the common ratio.
For the given sequence: 120, 40, 40/3, 40/9, 40/27, ... let's determine the explicit rule.
Identify the first term, a:
The first term is 120.
Determine the common ratio, r:
The ratio of any term to the previous term is 40 / 120, which simplifies to 1/3.
Write the explicit rule:
The nth term of a geometric sequence is given by the formula:
[tex]a(n) = a \times r^{n-1}[/tex]
Here, a = 120 and r = 1/3, so the explicit rule is:
[tex]a(n) = 120 \times (1/3)^{n-1}[/tex]
Complete question:
Enter the explicit rule for the geometric sequence. 120, 40, 40 / 3, 40 / 9, 40 / 27, ...
a(n)=
rewrite the radical in exponential form
Can ya help me I can give ya 20 points please help :)
Answer:
okkkkkkkkkkkkk
Step-by-step explanation:
Write the missing numbers to complete the pattern: 3, 6, 12 and so on
What is the Value of -3r+8 when r=4?
-12
-4
4
12
Answer:
The answer is -4
Step-by-step explanation:
-3x4+8 =-4
How to solve a linear equation
Step-by-step explanation:
Let's use this equation as an example - [tex]102=-7r+4[/tex]
The whole point of a linear equation is to find what the variable (which in our case is r ) equals to.
Our equation: [tex]102=-7r+4[/tex]
First step is to try to get r alone on one side.
First step: [tex]102-4=-7r[/tex] we wouldn't do [tex]102+4=-7r[/tex] (notice it's negative and not positive) since we're switching the positive 4 to the other side we'd switch the positive to a negative, making it [tex]102-4[/tex].
Know that everytime we switch a number or variable to the other side we must always switch the sign to the opposite of what it is. If it's dividing, we multiply, if we subtract, we add, etc. Same with the other way around.
Second step is to solve both sides of the equation.
Second step: [tex]98=-7r[/tex]
Third step is to divide the both sides by the number with the variable, making it 98 divided by -7r and -7r divided by -7
Third Step: [tex]\frac{98}{-7} =\frac{-7r}{-7}[/tex] = [tex]-14=r[/tex]
Now we've found our variable.
Answer: [tex]-14=r[/tex]
identify each coefficient.
1. 8x _
2. 3m _
3. y_
4. 14t _
The figures below are made out of circles, semicircles, quarter circles, and a square. Find the area and the perimeter of each figure and give your answers as a completely simplified exact value in terms of π (no approximations).
Answer:
A=16(pi-2),p=4(2 sqrt2+pi)
Step-by-step explanation: