Unit 5: Exponential and Logarithmic Functions Essay
Exponential Function
Exponential Functions:
An exponential equation is a type of transcendental equation, or equation that can be solved for one factor in terms of another. An exponential function f with base a is denoted by f (x) = ax, where a is greater than 0, a can not equal 1, and x is any real number. The base 1 is excluded because 1 to any power yields 1. For example, 1 to the fourth power is 1×1×1×1, which equals 1. That is a constant function which is not exponential, so 1 is not allowed to be the base of an exponential equation. Otherwise, the base of a can be any number that is greater than 0 and isn’t 1, and x can be any real number. The equation for the parent function of an exponential functions follows as so:
-Domain: (-, )
-Range: (0, )
-Intercept: (0,1)
-It increases
-x-axis is the horizontal asymptote
-Continous
-At x=1, y=a.
This equation shifts from the parent function based on the equation f(x) = k+a(x-h) . In this equation, k shifts the parent function vertically, up or down, depending on the value of k. The h value shifts the parent function to the left or right. If h equals 1, it goes to the right 1 unit, if it is negative 1, it goes to the left 1 unit. If a is negative, the parent function is reflected on the x-axis. If x is negative, the parent function is reflected on the y-axis.
In many applications, the natural base e is the most convenient base in an exponential equation. The value e is approximately 2.718281828. The natural base e works exactly like any other base. It is easy to think of e as a substitution for a in f (x) = ax. Its graph looks as so:
-Domain: (-, )
-Range: (0, )
-Intercept: (0,1)
-It increases
-x-a...
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... Association, if a the sound of a plane taking off is 1,000,000,000,000 times the threshold sound, and if the sound of a hand drill is 10,000,000,000 times the threshold sound, during which sound would you wear hearing protection?
Db(plane) = 10log[ I ÷ I0 ]
= 10log[ (1,000,000,000,000)I0 ÷ I0 ]
= 10log[1,000,000,000,000]
= 120 Decibels
Db(hand drill) = 10log[ I ÷ I0 ]
= 10log[ (10,000,000,000)I0 ÷ I0 ]
= 10log[10,000,000,000]
= 100 Decibels
Conclusion: If I am near a plane taking off, I should wear ear protection, and if I am near a hand drill that is on, I don’t need ear protection. This is because a plane taking off is 120 decibels, and the sound of a hand drill is 100 decibels, and 120 decibels is greater than 110 decibels, so you need protection, and 100 decibels is less than 110 decibels, so you don’t need ear protection.
In severe times such as the six miners missing by the Cave in at the Crandall Canyon coal mine in Utah, sound is an effective way to know the existence of the miners.
This called the Bronsted-Lowry Theory. This theory can be shown in the chemical reaction HCl+H2O -> Cl-+ H3O. The HCl gave up its hydrogen to the water.Furthermore, the products left over from reaction have their own names as well. The acid (HCl) having given up its hydrogen (Cl-) is called the conjugate base. The base (H2O) having received the hydrogen (H3O), is called the conjugate base. The last theory is, The Lewis Theory. In the Lewis Theory an acid is any substance in a chemical equation which accepts an electron pair and a base is any substance which gives away its electron pair. This theory is different because it is broad enough to include substances which do not include oxygen or hydrogen. An example of this can be seen in the reaction, BF3 + F− → BF4− BF3 is the acid and F− is the base. Outside of these definitions there are common properties of acids and bases. The both conduct electricity. Acids are sour in taste and turn litmas paper red. Bases are bitter in taste and turn litmas paper
of magnitudes is impossible, it is not possible to say by how much one exceeds
The Scholar: I think that's more a function of sound wave vibration than anything else.
Noise is ubiquitous in our environment. (Pediatrics , 1997) It is undesirable sound, unwanted sound. Sound is what we hear. It is vibration in a medium, usually air. Sound has intensity, frequency and duration. The ability to hear sounds at certain frequencies is more readily lost in response to noise. (Pediatrics , 1997). The further you are from sound the less effect you hear it but the more closer you are to sound the louder it is.
The Russian harlequin in Joseph Conrad’s Heart of Darkness expresses a common habit amongst sailors to smoke when he exclaims to Marlow, “Smoke? Where’s a sailor that does not smoke?” (132), correspondingly, in today’s world it is common for daily surroundings to include loud, obnoxious sounds that can potentially damage ones hearing permanently. This type of hearing disability is frequently referred to as N.I.H.L (Noise-Induced Hearing Loss). “N.I.H.L can be caused by a one-time exposure to an intense ‘impulse’ sound such as an explosion, or by continuous exposure to loud sounds over an extended period of time, such as noise generated in a woodworking shop” (“Noise-Indu...
Noise (Noise induced hearing loss NIHL): Another occupational hazard that contributes to workplace injury is excessive noise. Excessive noise may have adverse effects, which include, high blood pressure, stress, reduced performance and noise induced hearing loss. While there are other factors contributing to NIHL, the shortage of prevention is a high contributor. Engineering controls is one way of reducing noise at its source (Nelson et al. 20...
So now that we have the desired answer and know what the formula is, we just need to know what it means and why it works. So
Logarithms initially originated in an early form along of logarithm tables published by the Augustinian Monk Michael Stifel when he published ’Arithmetica integra’ in 1544. In the same publication, Stifel also became the first person to use the word ‘exponent’ and the first to indicate multiplication without the use of a symbol. In addition to mathematical findings, he also later anonymously published his prediction that at 8:00am on the 19th of October 1533, the world would end and it would be judgement day. However the Scottish astronomer, physicist, mathematician and astrologer John Napier is more famously known as the person who discovered them due to his work in 1614 called ‘Mirifici Logarithmorum Canonis Descriptio’. The reason they were created is to present and express numbers in a new form that was easy to work with. He was successful, as logarithms can be applied in many functions which are used commonly today. They were even more useful back around the time they were created however, as there were no calculators in existence. Scientists (astronomers in particular) had to do massive amounts of calculations on paper which was very time consuming and inconvenient. When logarithms were introduced to them, they weren’t obliged to spend so much time on tedious arithmetic. Logarithms are essentially just exponents, as they show values by using a base number that is raised to a given exponent. Stifel created his logarithm tables to change complicated multiplication and division problems into addition and subtraction equations.
“M.1HS.RBQ.6 create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. (Limit to linear and exponential equations, and, in the case of exponential equations, limit to situations requiring evaluation of exponential functions at integer inputs.) (CCSS.Math.Content.HSA-CED.A.2)” (West Virginia Department of Educaion, n.d., p. 2).
Chapter two of The Universe and the Teacup deals with exponential numbers. More precisely, it deals with the difficulty humans have in processing very large and very small numbers. The term the book uses to describe this difficulty is "number numbness."
P(Xn,Yn), on a curve, a line tangent to the curve at P crosses the X axis at a point whose X coordinate is closer to the root than Xn. This X coordinate, we will call Xn+1. Repeating this process using Xn+1 in place of Xn will return a new Xn+1 which will be closer to the root. Eventually, our Xn will equal our Xn+1. When this is the case, we have found a root of the equation. This method may be unnecessarily complex when we are solving a quadratic or cubic equation. However, the Newton-Raphson Method compensates for its complexity in its breadth. The following examples show the versatility of the Newton Raphson Method.
For a normal quadratic equation there is a well known formula to find the roots. There is a formula to find the roots of a 3rd and fourth degree equation but it can be troubling to find those roots, but if the function f is a polynomial of the 5th degree there is no formula that can enable us to find the root...
Any number written with one decimal point is called real constant or floating point constant.
in exponential form. For instance, in a base 2 system, 4 can be written as 2