...MAT220 119. Explain how to solve an exponential equation when both sides can be written as a power of the same base. When an exponential equation has both sides of the equation as the same base one needs to rewrite the equation in the form of bM=bN. For instance, 24x-3=8. To make this the same base we need to make 8 a base of two by writing it as 2^3. Then we have 24x-3=23. Then we get rid of the base and get 4x-3=3. Finally we solve for x. 4x-3=3 4x=6 x=23 120. Explain how to solve an exponential equation when both sides cannot be written as a power of the same base. Use 3x = 140 in your explanation. To solve this equation one needs to use a natural logarithm or ln. First take the ln of both sides, ln 3x= ln 140 Then using bx= x ln b, move the variable to the front, x ln 3 = ln 140 Solve for x, x= ln3ln140= 1.0986122887/4.9416424226 = 0.22231723680404. 121. Explain the differences between solving log31x - 12 = 4 and log31x - 12 = log3 4. When solving log31x - 12 = 4 one needs to write it in the form of bc=M. To do this we do the following; logbM=c means bc=M. 1) log31x - 12 = 4 2) 34=x-12 3) 81=x-12 4) x=93 In the case of log31x - 12 = log3 4, since the log is the same on both sides of the equation the will be omitted. The new equation would be; 1x-12=4. Then solve as normal. Add 12 to 4 to get 16, leaving 1x, which is just x and you have x=16. 122. In many states, a 17% risk of a car accident...
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...MA131 0 : Module 2 Exponential a nd Logarithmic Functions Exercise 2 .2 Solving Exponential and Logarithmic Equations 1 Answer the following questions to complete this exercise: 1. Solve the following exponential equation by expressing each side as a power of the same base and then equating exponents: 6 x = 216 2. Solve the following exponential equation: e x = 22.8 Express the solution in terms of natural logarithms. Then, use a calculator to obtain a decimal approximation for the solution. 3. Solve the following logarithmic equation: log 7 x = 2 Reject any value of x that is not in the domain of the original logarithmic expression. Give the exact answer. 4. Solve the following logarithmic equation: log ( x + 16) = log x + log 16 Reject any value of x that is not in the domain of the original logarithmic expression. Give the exact answer. 5. The population of the world has grown rapidly during the past century. As a result, heavy demands have been made on the world's resources. Exponential functions and equations are often used to model this rapid growth, and logarithms are used to model slower growth. The formula 0.0547 16.6 t Ae models the population of a US state, A , in millions, t years after 2000. a. What was the population in 2000? b. When will the population of the state reach 23.3 million? 6. The goal of our financial security depends on understanding how money in savings accounts grows in remarkable...
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...Nina Hills MAT 205 /Week 2 Focus on Application 07/11/2014 The concept of this week was to look at function problems that can include exponentials and logarithms with functions. These functions help with situations such as profit analysis, compound interest, continues compound interest or even doubling time for an investment. An example that I have that would go very well with today’s day in age would be simply the economy on its own. Our economy has taken such a huge turn downhill due to big banks making poor choices of investment. With that, many people don’t have savings accounts, 401K’s and such for their own future ahead. These two examples are examples of ways we may save for our retirement, but at this point there is a bare chance of that happening at an earlier on age. Many will have to work longer throughout their lives just to make sure that they are financially set when entering retirement. With the concepts of this week, we can calculate how long it would take to double a certain amount of investment in a certain time period with a fixed interest rate that would play upon a certain interval. A=P(1+r/m)^mt This equation can help determine t (time), for the principal to double. We can put in 2P for A, due to the other known values are r (interest rate) and m=1. Once we solve for t, we know the amount of time it will take to double our investment. With this week’s concept, we can predict at a pretty accurate rate the amount of time it takes to grow...
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...model a variety of realworld phenomena: growth of populations of people, animals, and bacteria; radioactive decay; epidemics; absorption of light as it passes through air, water, or glass; magnitudes of sounds and earthquakes. We consider applications in these areas plus many more in the sections very important. As a part of our BBA course, we are required to submit a term paper for every subject each semester. As our Advance Business Mathematics faculty Associate Professor Lt. Col. Md. Showkat Ali has asked us to submit a term paper on a topic upon our will. So, we have decided to choose “Exponential & Logarithmic Functions”. to graph exponential functions to evaluate functions with base e to learn the use of compound interest formulas to learn the changing from logarithmic to exponential form to learn the changing from exponential to logarithmic form to learn the evaluation of logarithms to learn the use of basic logarithmic properties to learn the use of graph logarithmic functions to find the domain of a logarithmic function to learn the use of common logarithms to learn the use of natural logarithms to learn the use of the product rule to learn the use of the quotient rule to learn the use of the power rule to...
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...This lab requires you to: • Evaluate exponential functions. • Graph exponential functions. • Evaluate functions with base e. • Change from logarithmic to exponential form. • Change from exponential to logarithmic form. • Evaluate logarithms. • Use basic logarithmic properties. • Graph logarithmic functions. • Find the domain of a logarithmic function. • Use common logarithms. • Use natural logarithms. • Use the product rule. • Use the quotient rule. • Use the power rule. • Expand logarithmic expressions. • Condense logarithmic expressions. • Use the change-of-base property. Answer the following questions to complete this lab: 1. State in a few words, what is an exponential function? 2. What is the natural exponential function? 3. Evaluate 4–1.5 using a calculator. Round your answer to three decimal places. 4. The formula S = C (1 + r)^t models inflation, where C = the value today r = the annual inflation rate S = the inflated value t years from now Use this formula to solve the following problem: If the inflation rate is 3%, how much will a house now worth $510,000 be worth in 5 years? 5. Write 6 = log2 64 in its equivalent exponential form. 6. Write 8y = 300 in its equivalent logarithmic form. 7. Hurricanes are some of the largest storms on earth. They are very low pressure areas with diameters of over 500 miles. The barometric air pressure in inches of mercury at a distance of x miles from the eye of a severe hurricane is modeled by the formula...
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...This is an essay about nothing in order to qualify for this site it must contain at least 250 words. So On the left-hand side above is the exponential statement "y = bx". On the right-hand side above, "logb(y) = x" is the equivalent logarithmic statement, which is pronounced "log-base-b of y equals x"; The value of the subscripted "b" is "the base of the logarithm", just as b is the base in the exponential expression "bx". And, just as the base b in an exponential is always positive and not equal to 1, so also the base b for a logarithm is always positive and not equal to 1. Whatever is inside the logarithm is called the "argument" of the log. Note that the base in both the exponential equation and the log equation (above) is "b", but that the x and y switch sides when you switch between the two equations.PrintHidden<p><font face="Arial" size="2" color="#000000">Note: The graphic in the box below is animated in the original ("live") web lesson.</font></p> —The Relationship Animated— | | If you can remember this relationship (that whatever had been the argument of the log becomes the "equals" and whatever had been the "equals" becomes the exponent in the exponential, and vice versa), then you shouldn't have too much trouble with logarithms. Copyright © Elizabeth Stapel 2002-2011 All Rights Reserved //(I coined the term "The Relationship" myself. You will not find it in your text, and your teachers and tutors will have no idea...
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...and brought the microscope over to my work area, making sure to carry the microscope by the arm and base. I uncovered and plugged in the microscope. I then went back to the cart and got a slide and slide cover, as well as a small glass bottle and dropper. I filled the small glass bottle with water and took everything back to my work area. I wrote a letter e on a piece of paper with a pen, pulled a strand of hair from my head and pulled a string off of my jacket. Then I turned on the microscope, prepared my slide and proceeded to look at each object under the microscope. Data: If the slide was too close or too far from the lens than you will not be able to see the specimen. The larger the magnification on the microscope the more detail that can be seen. The course and fine adjustment knobs move the slide up and down to help focus the specimen on the slide. The mechanical stage controls move the slide left and right, and forward and backwards. Findings: While observing the hair under the microscope I noticed that it is not smooth. The hair actually looks like it is made up of tiny scales. While observing the paper with the letter e written on it, I noticed that, just like the hair, the paper does not look smooth. The paper actually looks like a bunch of threads woven together like a birds nest. The ink on the paper only seemed to stick to the top layer or two of the paper material. I also observed that the letter appeared upside down and...
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...A Generalized Logarithm for Exponential-Linear Equations Dan Kalman Dan Kalman (kalman@email.cas.american.edu) joined the mathematics faculty at American University in 1993, following an eight year stint in the aerospace industry and earlier teaching positions in Wisconsin and South Dakota. He has won three MAA writing awards, is an Associate Editor of Mathematics Magazine, and served a term as Associate Executive Director of the MAA. His interests include matrix algebra, curriculum development, and interactive computer environments for exploring mathematics, especially using Mathwright software. How do you solve the equation 1.6x = 5054.4 − 122.35x? (1) We will refer to equations of this type, with an exponential expression on one side and a linear one on the other, as exponential-linear equations. Numerical approaches such as Newton’s method or bisection quickly lead to accurate approximate solutions of exponential-linear equations. But in terms of the elementary functions of calculus and college algebra, there is no analytic solution. One approach to remedying this situation is to introduce a special function designed to solve exponential-linear equations. Quadratic equations, by way of analogy, are √ solvable in terms of the special function x, which in turn is simply the inverse of a very special and simple quadratic function. Similarly, exponential equations are solvable in terms of the natural logarithm log, and that too is the inverse of...
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...Question 1 Evaluate the function at the indicated value of x. Round your result to three decimal places. Function: f(x) = 0.5x Value: x = 1.7 | | -0.308 | | | 1.7 | | | 0.308 | | | 0.5 | | | -1.7 | 5 points Question 2 Match the graph with its exponential function. | | y = 2-x - 3 | | | y = -2x + 3 | | | y = 2x + 3 | | | y = 2x - 3 | | | y = -2x - 3 | 5 points Question 3 Select the graph of the function. f(x) = 5x-1 | | | | | | | | | | | | | | | 5 points Question 4 Evaluate the function at the indicated value of x. Round your result to three decimal places. Function: f(x) = 500e0.05x Value: x=17 | | 1169.823 | | | 1369.823 | | | 1569.823 | | | 1269.823 | | | 1469.823 | 5 points Question 5 Use the One-to-One property to solve the equation for x. e3x+5 = 36 | | x = -1/3 | | | x2 = 6 | | | x = -3 | | | x = 1/3 | | | x = 3 | 5 points Question 6 Write the logarithmic equation in exponential form. log8 64 = 2 | | 648 = 2 | | | 82 = 16 | | | 82 = 88 | | | 82 = 64 | | | 864 = 2 | 5 points Question 7 Write the logarithmic equation in exponential form. log7 343 = 3 | | 7343 = 2 | | | 73 = 77 | | | 73 = 343 | | | 73 = 14 | | | 3437 = 2 | 5 points Question 8 Write the exponential equation in logarithmic form. 43 = 64 | | log64 4 = 3 | | | log4...
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...1. An exponential function is a function with a constant base that is changed by x, a variable. Exponential functions are used to predict changes in murder rates, bacteria growth even investments. This function can also be used in predicting rate of decay such as automobile value and radioactive half-life. 2. The natural exponential function, f(x) = ex, has a known base constant. Unlike other exponential functions where the constant, a, can be any real number, e is always 2.718. A good example of a natural exponential function is continuous compound interest. 3. Evaluate 4-1.5 = 0.125 4. Using the formula S = C(1 + r)t If the inflation rate is 3%, how much will a will a house now worth $510,000 be worth in five years? S = $510,000 ( 1 + .03 )5 S = $510,000 x 1.035 S = $591,229.78 5. Write 6 = log2 64 in its equivalent exponential form. y = loga x 6 = log2 64 x = ay 64 = 26 6. Write 8y = 300 in its equivalent logarithmic form. y = bx 300 = 8y logb (y) = x log8 (300) = y 7. Using the formula: f(x) = 0.48 In (x+1) + 27 a. Evaluate f(0) and f(100). Interpret the result. f(0) = 0.48in (1) + 27 = 27 says the barometric pressure at the eye is 27 f(100) = 0.48 (101) + 27 = 29.215 says the barometric pressure 100 miles from the eye is approximately 29.2 b. At what...
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...accurate b. As the percentage increases towards 100% the rate of kids that volunteer will slow and a large number of years will be needed to continue to approach 100% representative of a logarithmic function. An exponential function would reach 100% in a few years, which is not representative of the rate at which the kids are volunteering. 56. H=-[.521log2.521+.324log2.324+.0811log2.0811+.074log2.074] H=1.59 Section 4.5 6. 5x=13 Ln5x=13 Xln5=ln13 X = ln13/ln5 = 1.59 24. 5(1.2)3x-2 + 1 = 7 5(1.2)3x-2=6 ln1.23x-2=ln6/5 (3x-2)ln1.2 = ln6/5 3x-2 = ln(6/5)/ln1.2 3x = [ln(6/5)/ln1.2] +2 X = [[ln(6/5)/ln1.2] +2]/3 X=1 60. R=p-kln(t) r-p=-kln(t) p-r=kln(t) (p-r)/k = ln(t) e[(p-r)/k]=t 76. 20,000=16,000(1+r/4)5.25*4 1.25=(1+r/4)21 Ln1.25 = 21ln(1+r/4) Ln1.25/21 =ln(1+r/4) e.010626=1+r/4 1.0107 = 1+r/4 .0107=r/4 r=.04, therefore rate...
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...1. Read Module 3 Topic 7, Module 4 Topic 1 and 2, Module 5 Topic 1~2. 2. Do the drills for the topics. 3. Read the Chapter 3 sections 2, 5 and Chapter 4 sections 1~3 in your textbook. 4. Do Homework for week 5 (you can find the list in the conference). Week 5 Supplementary Notes Chapter 3 Section 3.2: Polynomial Function of Higher Degree A polynomial function P is given by , where the coefficients are real numbers and the exponents are whole numbers. This polynomial is of nth degree. Far-Left and Far-Right Behavior The behavior of the graph of a polynomial function as x becomes very large or very small is referred to as the end behavior of the graph. The leading term of a polynomial function determines its end behavior. x becomes very large x → ∞ x becomes very large x → ∞ x becomes very small -∞ ← x x becomes very small -∞ ← x We can summarize the end behavior as follows: The Leading-Term Test If is the leading term of a polynomial, then the behavior of the graph as x → ∞ or as x → −∞ can be described in one of the four following ways. If n is even and an >0: ▼ ▼ | If n is even and an <0:▲ ▲ | If n is odd and an >0: ▲▼ | If n is odd and an <0: ▲ ▼ | Polynomial Function, Real Zeros, Graphs, and Factors (x − c) If c is a real zero of a function (that is, f(c)=0), then (c,0) is an x-intercept of the graph...
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...Writing in Mathematics Exercises 119. Explain how to solve an exponential equation when both sides can be written as a power of the same base. a. An exponential equation is defined as an equation that contains a variable in an exponent. In order to solve an exponential equation we need to look at the steps that are required. Exponential equations that have the same base are in the form of If bm=bn. When we see an equation of exponents with the same base we will find the answer by setting the exponents equal to each other. The formula that we can look at to understand how to solve an exponential equations is defined as If bm=bn, then m=n. The steps that we will take are as follows: 1. Rewrite the equation in the form bm=bn. 2. Set m=n. 3. Solve for the variable. 120. Explain how to solve an exponential equation when both sides cannot be written as a power of the same base. Use 3x=140 in your explanation. a. In order to solve an exponential equation when both sides are not written with the same base, we need to use logarithms. To convert an exponential equation into logarithmic form we look at the formulas below: by=x is equivalent to y=logbx Using 3x=140, we would solve the problem by performing the following steps: 1. Isolate the exponential equation 2. Take the natural logarithm on both sides of the equation for bases other than 10. Take the common logarithm on both sides of the equation for base 10. 3. Simplify using one of the following...
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...Speed-Accuracy Tradeoff Mikko Allen D. San Miguel Pamantasan ng Lungsod ng Maynila Abstract In Speed-Accuracy Tradeoff, a low speed means higher accuracy and a high speed means lower accuracy. This principle is applied in terms of reading. It is predicted that when people read faster, they tend to be less accurate about what they read. This is tested by compelling subjects, to read faster. The subjects were asked to read faster than average, by increasing the target number of lines they were required to read. Also, they were asked to cross-out all the letter e’s that they see as they go through a reading material. Most of the results are consistent with the prediction. When subjects read faster, they committed more mistakes and when they read slower, they were able to commit less. However, other findings are inconsistent with this prediction. For example, even when the subjects were not required to read faster than average, they were still unable to gain a higher score in correctly crossing-out the letter e’s that they came across. Introduction Is a job done fast, a job done inaccurately or can a job be done both fast and accurate? Individuals attempt to perform well in both factors (Zimmerman, 2011). Everyone wants a job well done in a minimum amount of time. For example, it would be amazing if a repairman can fix a broken phone properly in less than the average time it usually takes; this is beneficial to both parties. The client saves more time than usual. Similarly...
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...“anyone lived in a little how town” ee cummings • non-conformity • form = function o Poe’s unity of effect o poem = unique, does not conform to any poetic standards, grammatical rules, expectations • songs: o Justice & Independence, Jack & Diane (JC Mellencamp) o The Dance, The River (Garth Brooks) E. E. Cummings' "anyone lived in a pretty how town" tells the story of anyone. The name has a double meaning; anyone could be anyone in the dictionary definition sense, and could be seen as a singular entity, reinforcing the theme of isolation the independent individual has from the rest of society. The events all occur in a "pretty how town". "Pretty" connotes a mere façade, describing the superficiality of the town's inhabitants. "How", an adverb, is used as an adjective here. It could be describing the extent of the town's prettiness, but a better reason is that it describes the routine humdrum of the town's activities, since "how" also means "in a method or manner". The juxtapositions continue into the next line, "(with up so floating many bells down)". The rhythm of the line and the vowels emulate both the motion and the sounds of bells. This line occurs again later in the poem, and its function here is the same as it is there - to signify the passing of time. The next line is an ordered list of the seasons, also symbolizing the passing of time, describing anyone's activities as occurring continuously. The activities...
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