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Exponential & Logarithmic Function

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Most of the functions we have studied have been polynomial or rational functions, with a few others involving roots of polynomial or rational functions. Functions that can be expressed in terms of addition, subtraction, multiplication, division, and the taking of roots of variables and constants are called algebraic functions.

In exponential & logarithmic functions we introduce and investigate the properties of exponential functions and Logarithmic functions. These functions are not algebraic; they belong to the class of transcendental functions. Exponential and logarithmic functions are used to 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”.

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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 learn the use of Expanding logarithmic expressions to learn the use of Condensing logarithmic expressions to learn the use of the change of base property to learn the use of like bases to solve exponential equations. to learn the use of logarithms to solve exponential to learn the use of the definition of a logarithm to solve logarithmic equations to solve applied problems involving exponential and logarithmic equations to model exponential growth and decay to complete a requirement of the course

We had enough scope to make this term paper. We have tried to cover as much as detail as possible. We checked every possible aspect to widen and perfectly use our scope.

We have tried our best to submit this Term Paper according to the requirements and according to our ability. There may be a few mistakes, because we are still novice in this line of work. Moreover, we don’t have the vast knowledge on this topic. That is why there may be a few mistakes for being rookies.

The equation f(x) = bx b> 0 b ≠ 1

Defines an exponential function for each different constant b, called the base. The independent variable x may assume any real value. Thus the domain of f is the set of all real numbers and the range of f is the set of all positive real numbers.

Let f(x) = bx be an exponential function, b>0, b1. Then the graph of f(x): 1. Is continuous for all real numbers 2. Has no sharp corners 3. Passes through the point (0, 1) 4. Lies above the x axis, which is a horizontal asymptote 5. Increases as x increases if b>1; decreases as x increases if 0 0 as x > . The x axis is a horizontal asymptote, 1/2 is the y intercept, and there is no x intercept. We plot the intercept and some additional points and sketch the graph of g

Exponential functions whose domains include irrational numbers obey the familiar laws of exponents for rational exponents. We summarize these exponent laws here and add two other important and useful properties.

Property 2 is another way to express the fact the exponential function f (x) = ax is oneto-one. Because all exponential functions pass through the point (0, 1), property 3 indicates that the graphs of exponential functions with different bases do not intersect at any other points.

Surprisingly, among the exponential functions it is not the function g(x) = 2x with base 2 or the function h(x) = 10x with base 10 that is used most frequently in mathematics. Instead, it is the function f (x) = ex with base e, where e is the limit of the expression

as x gets larger and larger So, for x a real number, the equation f(x) = ex defines the exponential function with base e.

If you deposit $5,000 in an account paying 9% compounded daily,* how much will you have in the account in 5 years? Compute the answer to the nearest cent.

If $100 is invested at an annual rate of 8% compounded continuously, what amount, to the nearest cent, will be in the account after 2 years?

Populations tend to grow exponentially and at different rates. A convenient and easily understood measure of growth rate is the doubling time—that is, the time it takes for a population to double. Over short periods the doubling time growth model is often used to model population growth:

and the population is double the original, as it should be. We use this model to solve a population growth problem in Example 1.

Mirpur - 1 has a population of approximately 6 million and it is estimated that the population will double in 36 years. If population growth continues at the same rate, what will be the population? (A) 15 years from now? (B) 40 years from now?

Cholera, an intestinal disease, is caused by a cholera bacterium that multiplies exponentially by cell division as modeled by

where N is the number of bacteria present after t hours and N 0 is the number of bacteria present at t = 0. If we start with 1 bacterium, how many bacteria will be present in (A) 5 hours? (B) 12 hours?

The radioactive isotope gallium 67 (67 Ga), used in the diagnosis of malignant tumors, has a biological half-life of 46.5 hours. If we start with 100 milligrams of the isotope, how many milligrams will be left after (A) 24 hours? (B) 1 week?

The logarithm of a number is the exponent to which a fixed number, the base, has to be raised to produce that number. For example, the logarithm of 1000 to base 10 is 3, because 1000 is 10 to the power 3: 1000 = 103 = 10 × 10 × 10. More generally, if x = by, then y is the logarithm of x to base b, and is written logb(x), so log10 (1000) = 3

We now look into the matter of converting logarithmic forms to equivalent exponential forms, and vice versa.

Change each logarithmic form to an equivalent exponential form.

Change each exponential form to an equivalent logarithmic form.

Solutions of the Equation y = log b x

Find x, b, or y as indicated.

Using Logarithmic Properties

Simplify, using the properties

John Napier (1550–1617) is credited with the invention of logarithms, which evolved out of an interest in reducing the computational strain in research in astronomy. This new computational tool was immediately accepted by the scientific world. Now, with the availability of inexpensive calculators, logarithms have lost most of their importance as a computational device. However, the logarithmic concept has been greatly generalized since its conception, and logarithmic functions are used widely in both theoretical and applied sciences. Of all possible logarithmic bases, the base e and the base 10 are used almost exclusively. To use logarithms in certain practical problems, we need to be able to approximate the logarithm of any positive number to either base 10 or base e. And conversely, if we are given the logarithm of a number to base 10 or base e, we need to be able to approximate the number. Historically, tables were used for this purpose, but now calculators are used because they are faster and can find far more values than any table can possibly include. Common logarithms, also called Briggsian logarithms, are logarithms with base 10. Natural logarithms, also called Napierian logarithms, are logarithms with base e. Most calculators have a function key labeled “log” and a function key labeled “ln.” The former represents the common logarithmic function and the latter the natural logarithmic function. In fact, “log” and “ln” are both used extensively in mathematical literature, and whenever you see either used in this book without a base indicated, they should be interpreted as in the box.

Evaluating a Base 3 Logarithm

Evaluate log3 5.2 to four decimal places.

The human ear is able to hear sound over an incredible range of intensities. The loudest sound a healthy person can hear without damage to the eardrum has intensity 1 trillion (1,000,000,000,000) times that of the softest sound a person can hear. Working directly with numbers over such a wide range is very cumbersome. Because the logarithm, with base greater than 1, of a number increases much more slowly than the number itself, logarithms are often used to create more convenient compressed scales. The decibel scale for sound intensity is an example of such a scale. The decibel, named after the inventor of the telephone, Alexander Graham Bell (1847–1922), is defined as follows:

Sound Intensity

Find the number of decibels from a whisper with sound intensity 5.2 x 10-10 watts per square meter. Compute the answer to two decimal places.

The energy released by the largest earthquake recorded, measured in joules, is about 100 billion (100,000,000,000) times the energy released by a small earthquake that is barely felt. Over the past 150 years several people from various countries have devised different types of measures of earthquake magnitudes so that their severity could be easily compared. In 1935 the California seismologist Charles Richter devised a logarithmic scale that bears his name and is still widely used in the United States. The magnitude M on the Richter scale* is given as follows:

where E is the energy released by the earthquake, measured in joules, and E is the energy released by a very small reference earthquake, which has been standardized to be

The 1906 San Francisco earthquake released approximately 5.96 x 1016 joules of energy. What was its magnitude on the Richter scale? Compute the answer to two decimal places.

If the energy release of one earthquake is 1,000 times that of another, how much larger is the Richter scale reading of the larger than the smaller?

Thus, an earthquake with 1,000 times the energy of another has a Richter scale reading of 2 more than the other.

Equations involving exponential and logarithmic functions, for example

are called exponential and logarithmic equations, respectively. We solve such equations to find the x intercepts of a function, or more generally, to find where the graphs of two functions intersect. Logarithmic properties play a central role in the solution of both exponential and logarithmic equations.

Finding x Intercepts

Find the x intercept(s) of f(x) = 23x-2 - 5 to four decimal places.

Compound Interest

Solving a Logarithmic Equation

Solve log (x + 3) + log x = 1, and check.

Finally, we can say that we have covered every basic possible thing of exponential & logarithmic functions. We now know many mathematical problems of exponential & logarithmic functions. We now know the use of compound interest formulas the changing from logarithmic to exponential form, use of common logarithms, the use of natural logarithms, the use of the product rule, the use of the quotient rule, the use of the power rule and many other things.

College Algebra; Barnet, Raymond A, Zeigler, Michael R, Bayleen, Karl E, 7th edition.

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