For any positive number b not equal to 1 and any number or variable n, evaluate the following expression.
logb(b^n) =
with logb being the base
Answer:
[tex]log_{b} b^{n}[/tex] = n.
Step-by-step explanation:
Given : [tex]log_{b} b^{n}[/tex] , b is any positive number not equal to 1 and n is any number.
To find : Evaluate the expression [tex]log_{b} b^{n}[/tex].
Formula used : [tex]log_{b} x^{y}[/tex]. = y ∙ [tex]log_{b}[/tex](x) and
[tex]log_{b}(b) = 1.
Solution : We have [tex]log_{b} b^{n}[/tex].
By logarithm rule : [tex]log_{b} x^{y}[/tex]. = y ∙ [tex]log_{b}[/tex](x).
Then [tex]log_{b} b^{n}[/tex] = n∙ [tex]log_{b}[/tex](b).
By logarithm rule : [tex]log_{b}(b) = 1.
Now, n∙ [tex]log_{b}[/tex](b) = n.
Therefore, [tex]log_{b} b^{n}[/tex] = n.
Estimate the volume of material in a cylindrical shell with height 30 in., radius 2 in., and shell thickness 0.5 in.
Water is pumped out of a holding tank at a rate of 6-6e^-0.13t liters/minute, where t is in minutes since the pump is started. If the holding tank contains 1000 liters of water when the pump is started, how much water does it hold one hour later?
out of a swarm of bees one fifth part settled on a kadamba blossom; one third on a silihindra flower; three times the difference of those numbers flew to the bloom of a kutaja. one bee, which remained, hovered about in the air. tell me, charming woman, the number of bees
What is the sum of square root -2 and squareroot -18
The sum of the square roots of -2 and -18 is [tex]\( 4\sqrt{2}i \)[/tex], where i is the imaginary unit.
The sum of the square roots of -2 and -18 involves complex numbers because both -2 and -18 are negative.
The square root of a negative number is not a real number, but it can be expressed using imaginary units.
The square root of -2 can be represented as [tex]\( \sqrt{-2} = \sqrt{2}i \),[/tex] where i is the imaginary unit, defined as [tex]\( i = \sqrt{-1} \).[/tex]
Similarly, the square root of -18 can be written as [tex]\( \sqrt{-18} = \sqrt{18}i \).[/tex]
Now, to find the sum of these square roots:
[tex]\[ \sqrt{-2} + \sqrt{-18} = \sqrt{2}i + \sqrt{18}i \][/tex]
[tex]\[ = \sqrt{2}i + \sqrt{2 \times 9}i \][/tex]
[tex]\[ = \sqrt{2}i + 3\sqrt{2}i \]\\\\ = (\sqrt{2} + 3\sqrt{2})i \][/tex]
[tex]\[ = 4\sqrt{2}i \][/tex]
So, the sum of[tex]\( \sqrt{-2} \) and \( \sqrt{-18} \)[/tex] is [tex]\( 4\sqrt{2}i \).[/tex]
how many solutions are there to this equation?
9x - 16 = 9x - 23
a. infinitely many
b. 0
c. 1
Answer: the correct option is
(b) 0.
Step-by-step explanation: We are given to find the number of solutions for the following linear equation :
[tex]9x-16=9x-23~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(i)[/tex]
To find the number of solutions of the given equation, we must try to solve the equation and find the value of x.
From equation (i), we have
[tex]9x-16=9x-23\\\\\Rightarrow 9x-9x=-23+16\\\\\Rightarrow 0=-7,[/tex]
which can never be possible.
Thus, the given equation has NO solution.
Option (b) is CORRECT.
round 624 to the nearest hundred
Help me please !
Scores on a statewide standardized test are normally distributed with a mean of 12.89 and a standard deviation of 1.95. Certificates are given to students whose scores are in the top 2% of those who took the test. This means that they scored better than 98% of the other test takers. Marcus received his score of 13.7 on the exam and is wondering why he didn’t receive a certificate. Show all work to determine whether Marcus’ score was high enough to earn a certificate. Write a letter to Marcus explaining whether or not he will be receiving a certificate. Include a brief summary of your statistical analysis in your letter.
When sam breathes in, his chest expands, causing air to fill his lungs. which structure causes his chest to expand?
Write an equation of the direct variation that includes the point (9, –12).
(A.) y = 1 1/3x (B.) y = - 1/12x (C.) y = 1 1/3x (D.) -3/4x ...?
The equation of direct variation for the point (9, -12) would be y = -1 1/3x. The constant of variation, 'k', is found by dividing 'y' by 'x'. Therefore, 'k' is -12/9, which simplifies to -1 1/3.
Explanation:In a direct variation, the equation can be represented in the form y = kx, where 'k' is the constant of variation. Here, 'k' can be found by dividing 'y' by 'x'. So, in the given point (9, -12), 'k' would be -12/9, which simplifies to -4/3 or -1 1/3. Therefore, the equation of the direct variation that includes the point (9, -12) would be y = -1 1/3x. The provided options don't include this equation, so it might be a mistake in the question options. Remember, the correct equation should always have 'y' and 'x' directly vary, and 'k' have the same sign as the given 'y' value.
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Which expression is equaivalent to mn+z
nm+n
z+mz
mz+n
z+nm
Answer:
D. z+nm
Step-by-step explanation:
Equaivalent to mn+z: Just switch the letters around and you get z+nm
appleseed a tech company spends a fixed cost of $1500 to manufacture the wristbands for their new appleseed watch plus $100 per unit manufactured solution
a. The slope is 100. It shows the cost per wristband.
b. The y-intercept is 1500. It shows the fixed cost for manufacturing the wristbands.
Given that,
The fixed cost of $1,500 for manufacturing the wristbands. And, there is $100 per unit.Based on the above information, we can conclude that
a. The slope is 100. It shows the cost per wristband.
b. The y-intercept is 1500. It shows the fixed cost for manufacturing the wristbands.
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Solve the system by the elimination method. Check your work. 2a + 3b = 6 5a + 2b - 4 = 0
Answer:
Step-by-step explanation:
a=0
b=2
answer (0,2)
What is 7.58E8 expressed in standard notation?
If xy^2 = 20, and x is decreasing at the rate of 3 units per second, the rate at which y is changing when y = 2 is nearest to?
What is the median of this set of data? 66, 51, 77, 68, 60, 75, 54, 80
67````````````````````````
100 plus the product of a number and -2 equals 50 What is the number?
What is the median distance jumped 7.48, 7.60, 7.15, 7.44, 7.73, 7.72
carla looks from the top of a diving board, and lines up a ball floating on the surface of the water with the bottom of the opposite corner of the pool. calculate the total distance across the surface of the pool. explain your reasoning
If two angles of a triangle measure 43 degrees and 48 degrees the triangle is
The third angle for the triangle is equal to 89°.
What is geometry?Geometry is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is called a geometer.
A triangle is a three-edged polygon with three vertices. It is a fundamental shape in geometry. Triangle ABC denotes a triangle with vertices A, B, and C.
Given that the two angles of a triangle measure 43 degrees and 48 degrees. the value of the third angle will be calculated as,
Third angle = 180 - 43 - 48
Third angle = 89°
The complete question is given below.
If two angles of a triangle measure 43 degrees and 48 degrees what is the third angle of the triangle?
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What is the average rate of change of the function f(x)=5(2)^x from x = 1 to x = 5?
Your iron works has been contracted to design and build a 500-ft3 square-based, open-top, rectangular steel holding tank for a paper company. The tank is to be made by welding half-inch-thick stainless steel plates together along their edges. As the production engineer, your job is to find dimensions for the base and height that will make the tank weight as little as possible. What dimensions do you tell the shop to use?
To minimize the weight of the tank, the base length should be (4/3) times the height. The height should be (3/4)^(1/4) times the fourth root of (2V/π).
Explanation:To find the dimensions that will make the tank weight as little as possible, we need to consider the volume and weight of the tank. Since the tank is square-based, the volume can be calculated by V = bh^2, where b is the length of one side of the base and h is the height.
The weight of the tank can be calculated by multiplying the volume by the density of the stainless steel. W = V * density. Since we want to minimize the weight, we want to minimize the volume.
To do this, we can differentiate the volume equation with respect to b and set it equal to zero. Solving for b, we find that b = (4/3) * h. Substituting this value into the volume equation gives: V = (4/3) * h^3.
To further minimize the volume, we can take the derivative of the volume equation with respect to h and set it equal to zero. Solving for h, we find that h = (3/4)^(1/4) * (2V/π)^(1/4).
Calculate the average rate of change for the function f(x) = −x4 + 4x3 − 2x2 + x + 1, from x = 0 to x = 1.
0
1
2
7
Answer:
I believe the answer is (C.2)
Step-by-step explanation:
The answer isn’t 1. I took the test and ‘B’ is incorrect
Final answer:
C: 2
To find the average rate of change of the function f(x) from x = 0 to x = 1, we compute (f(1) - f(0)) / (1 - 0), which results in (3 - 1) / 1 = 2. The average rate of change is therefore 2.
Explanation:
To calculate the average rate of change for the function f(x) = −x⁴ + 4x³ − 2x² + x + 1 from x = 0 to x = 1, we use the formula:
Average rate of change = ∆f/∆x = (f(x_2) - f(x_1)) / (x_2 - x_1)
First, we need to find the values of f(x) at x = 0 and x = 1:
f(0) = −(0)⁴ + 4(0)³ − 2(0)² + (0) + 1 = 1f(1) = −(1)⁴ + 4(1)³ − 2(1)² + (1) + 1 = 3Using these values, we calculate the average rate of change:
Average rate of change = (3 - 1) / (1 - 0) = 2 / 1 = 2
Therefore, the average rate of change of the function from x = 0 to x = 1 is 2.
Ray is 2 inch taller than Lin, and Ray is 3 inch taller than Sam. What is the average height of ray, Lin and Sam? The median height of Ray, Lin and Sam?
Harita must memorize 90 measures of music for her cello solo at a concert. She plans on memorizing 18 new measures for every 3 days of practice. Which equation can be used to determine m, the number of measures Harita still needs to memorize, as a function of d, the number of days of practice since she began learning the piece?
Answer:
m=90-6 days or b
Step-by-step explanation:
i got a hundred on the quiz...
The scores on a final exam were approximately normally distributed with a mean of 82 and a standard deviation of 11. If 85 students took the exam, and above a 60 is a passing grade, how many students failed the exam?
A. 13
B. 1
C. 2
D. 12
Answer:
2 students failed in the final exam.
Step-by-step explanation:
The scores on a final exam were approximately normally distributed.
We know that,
[tex]Z=\dfrac{X-\mu}{\sigma}[/tex]
X = raw score = 60
μ = mean = 82
σ = standard deviation = 11
Putting the values,
[tex]Z=\dfrac{60-82}{11}=-2[/tex]
From Normal distribution table, we get
[tex]P(-2)=0.0227=2.27\%[/tex]
Hence, 2.27% of 85 students failed the final exam.
So the number of students who failed the exam is,
[tex]=85\times 0.0227=1.9295\approx 2[/tex]
A number decreased by 10 is greater than -5.
How can I write this in an inequality?
To solve a three-variable system of equations, you can use a combination of the elimination and substitution mehtods.
True or False ...?
Final answer:
True, a combination of the elimination and substitution methods can solve a three-variable system of equations.
Explanation:
The statement is True. We can use a combination of the elimination and substitution methods to solve a three-variable system of equations. In the elimination method, we aim to eliminate one variable by adding or subtracting the equations, allowing us to solve for the remaining variables. This method works well when the coefficients of one variable are opposites in two equations.
In the substitution method, we solve one equation for one variable and substitute this expression into the other equations. This allows us to solve for the remaining variables. This method is more straightforward when one equation can be easily solved for one variable.
Expression as a decimal number 6x0.1 6x0.01 1x0.001
Final answer:
To express the given multiplications as decimal numbers: 6 times 0.1 equals 0.6, 6 times 0.01 equals 0.06, and 1 times 0.001 equals 0.001. Moving the decimal point due to multiplication by powers of 10 is the underlying concept.
Explanation:
The student's question involves expressing multiplications involving decimals in decimal form. To convert each term:
6 x 0.1 is moving the decimal point one place to the left in the number 6, which results in 0.6.6 x 0.01 is moving the decimal point two places to the left in the number 6, giving 0.06.1 x 0.001 is moving the decimal point three places to the left in the number 1, resulting in 0.001.When numbers are multiplied by powers of 10, the decimal point is shifted to the right for positive exponents and to the left for negative exponents. Scientific notation is often used for this purpose.
Here are examples demonstrating the use of scientific notation:
0.006 can be expressed as 6 x 10-3 (moving the decimal point three places to the left).0.0000045 becomes 4.5 x 10-6 (the decimal point moves six places to the left).Negative exponents indicate the decimal point moves to the left, whereas positive exponents indicate it moves to the right. For instance, 2.4 x 10-2 is equivalent to 0.024 as the decimal has moved two places to the left.
Kayla notices that she can buy snacks and candy in bulk at the grocery store. The snacks she wants to buy cost $1.29 per kilogram.
Leslie can determine which option will give her more yogurt by comparing the weights of the two options.
Explanation:To determine which option will give Leslie more yogurt, she needs to compare the weights of the two options. First, she needs to convert the weight of the first option from pounds to ounces. Since 1 pound is equal to 16 ounces, the 2 pound container is equal to 32 ounces. Next, she can compare the weights of the two options. The first option is 32 ounces and the second option is 5 containers of 6 ounces each, which is a total of 30 ounces. Therefore, the first option will give Leslie more yogurt.