Exponential And Logarithmic Models Ppt To Pdf

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Exponential-and-Logarithmic-Functions_0.ppt

Exponential Functions 1 Define an exponential function. Itsdomainis ,. Findg 2. Leth x. Find F 3. Graph the exponential function f x 3.

Plot the points and draw a smooth curve. As x increases in the positive direction, y decreases towards 0. The graph of is a smooth and continuous curve, and it passes through the points. Use transformations to sketch the graph of each function. State the domain and range of each function and the horizontal asymptote of its graph. It is an application of exponential growth. Interest that is compounded annually is paid once a year. Amount after t time periods Number of time periods.

Only n, the number of times interest is compounded each year, changes. A technician to the French microbiologist Louis Pasteur noticed that a certain culture of bacteria in milk doubles every hour.

If the bacteria count B t is modeled by the equation B t 2 t , with t in hours, find a. Solution a. Initial size B0 B 0 20 1 This irrational number is denoted by e and is sometimes called the Euler number. Convert eight years and three months to 8. Solution continued b. With interest compounded quarterly, 4t 4 r 0. Solution continued d. With interest compounded continuously, 0. Notice the dramatic difference between quarterly and continuous compounding and the dramatic difference between simple interest and compound interest.

In the year , the human population of the world was approximately 6 billion and the annual rate of growth was about 2. Using the model on the previous slide, estimate the population of the world in the following years.

The model predicts that if the rate of growth is 2. The model predicts that the world had over 4. The actual population in was 5.

The logarithmic function is the inverse function of the exponential function. Inverse Functions Certain pairs of one-to-one functions undo one another. Inverse Functions As further examples, check that f 3 29 and g 29 3,. Because of this property, g is called the inverse of. Inverse Function Let be a one-to-one function. Then g is the inverse function of if f o g x x for every x in the domain of g, and g o f x x for every x in the domain of.

Find the value of each of the following logarithms. Solution continued y 1 c. Solve each equation. Solution continued 1 b.

Applying the basic properties of logarithms 1. Logarithms of 0 and negative numbers are not defined. The domain of f is , 2. Solution by plotting points Method 1 Make a table of values. Domain , Domain 0, Range 0, Range , 2.

The graph is a The graph is a continuous smooth continuous smooth curve that passes curve that passes through the points through the points 1 1 1, , 0, 1 , and , 1, 1, 0 , and a a 1, a.

State the domain and range and the vertical asymptote for the graph of each function. Sketch the graph of y 2 log x 2. Replacing x with x 2 shifts the graph two units right.

Solution continued Step 2: Multiplying Step 3: Adding 2 by 1 reflects the graph shifts the graph in the x-axis. Write each expression in expanded form. Solution continued c. Solution continued 1 d. Let a, b, and x be positive real numbers with a 1 and b 1. Compute log by changing to a. Solution log13 a.

Evaluate each expression. How long will it take to double your money if it earns 6. At what rate of return, compounded continuously, would your money double in 5 years? It will take 11 years to double your money. Your investment will double in 5 years at the rate of Give the solution to the nearest thousandth. Solution While any appropriate base b can be used, the best practical base is base 10 or base e.

We choose base e natural logarithms here. Solution 7 x 12 In 7 In 12 x Property of logarithms. Solution 32 x 1. Solution Write the terms 2 x In 3 x In. Solution In. Give solutions to the nearest thousandth. Solution x In Square root property. Solution 3 x x 2 10 2 Multiply. Solution 1 The number 6 is negative, so x 1 is negative. Therefore, log x 1 is not defined and this proposed solution must be discarded.

The local McDonalds franchise has discovered that when coffee is poured from a coffeemaker whose contents are F into a noninsulated pot, after 1 minute, the coffee cools to F if the room temperature is 72F. How long should the employees wait before pouring the coffee from this noninsulated pot into cups to deliver it to customers at F?

Solution continued Substitute this value for k. A is the quantity after time t. A chemical spill deposits 60, cubic meters of soluble toxic waste into a large lake.

Solution continued 1. Find A0. So A 60,e rt. Find r. Open navigation menu. Close suggestions Search Search. User Settings. Skip carousel. Carousel Previous. Carousel Next. What is Scribd? Uploaded by Rommel Cruz. Document Information click to expand document information Date uploaded Aug 16, Did you find this document useful?

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Assessment: Presentation on Exponential Functions, Day 1 of 2

Exponential Functions 1 Define an exponential function. Itsdomainis ,. Findg 2. Leth x. Find F 3. Graph the exponential function f x 3. Plot the points and draw a smooth curve.

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Mathematical Modeling with Exponential and Logarithmic Functions. Learning Objective s. You have seen many different kind of functions. You know that each one can be used to model some kind of situation in the real world. Exponential and logarithmic functions are no exception! Using Logarithmic Functions.

6.5: Applications of Exponential and Logarithmic Functions

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Applications of Exponential Functions

Graph the relation in blue. For equations containing logarithms, properties of logarithms may not always be helpful unless the variable is inside the logarithm. Square all logarithmic expressions and solve the resulting quadratic equation. There are 6 snakes in a certain valley. Move the exponent out front which turns this into a multiplication problem. The Integration Guidelines 1.

Он делает то, на что запрограммирован, а потом исчезает. Фонтейн сурово смотрел на Джаббу: - И на что же запрограммирован этот червяк. - Понятия не имею, - сказал Джабба.  - Пока он ползет и присасывается к нашей секретной информации. После этого он способен на. Он может стереть все файлы, или же ему придет в голову напечатать улыбающиеся рожицы на документах Белого дома. Голос Фонтейна по-прежнему звучал спокойно, деловито: - Можете ли вы его остановить.

logarithmic and exponential

Сьюзан заглянула в распечатку через плечо Джаббы.

Однажды вечером на университетском представлении Щелкунчика Сьюзан предложила Дэвиду вскрыть шифр, который можно было отнести к числу базовых. Весь антракт он просидел с ручкой в руке, ломая голову над посланием из одиннадцати букв: HL FKZC VD LDS В конце концов, когда уже гасли огни перед началом второго акта, его осенило. Шифруя послание, Сьюзан просто заменила в нем каждую букву на предшествующую ей алфавите.

 Echo un poco de Smirnoff? - настаивал бармен.  - Плеснуть чуточку водки. - No, gracias. - Gratis? - по-прежнему увещевал бармен.  - За счет заведения.

Но Сьюзан его не слушала. Она была убеждена, что должно найтись какое-то другое объяснение. Сбой. Вирус.

Его слова не сразу дошли до ее сознания. Стратмор убил Чатрукьяна. Хейл, видимо, не догадывается, что она видела его внизу. - Стратмор знает, что я это видел! - Хейл сплюнул.

УМБРА, что было высшим уровнем секретности в стране. Никогда еще государственные секреты США не были так хорошо защищены. В этой недоступной для посторонних базе данных хранились чертежи ультрасовременного оружия, списки подлежащих охране свидетелей, данные полевых агентов, подробные предложения по разработке тайных операций. Перечень этой бесценной информации был нескончаем.

Есть шанс, что его партнер пока ничего не знает. Испанские власти обещали придержать информацию - столько, сколько смогут. Мы узнали об этом лишь благодаря оперативности КОМИНТа.  - Стратмор внимательно посмотрел не.  - Я должен найти его партнера, прежде чем он узнает о смерти Танкадо.

Он слишком долго обделял. И ради. Чтобы увидеть, как какой-то молодой профессор украл его мечту. Стратмор холил и лелеял Сьюзан, оберегал. Он заслужил .

Халохот ошибся с местом действия. Быть может, смерть Танкадо в публичном месте была необходимостью, однако публика возникла чересчур. Халохот был вынужден скрыться, не успев обыскать убитого, найти ключ. А когда пыль осела, тело Танкадо попало в руки местной полиции.

Наконец канадец опомнился. - Из консульства? - Его тон заметно смягчился. Беккер кивнул.

Через несколько секунд двенадцатитонная стальная махина начала поворачиваться. Она попыталась собраться с мыслями, но они упрямо возвращали ее к .

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    Exponential Functions; Logarithmic Functions; Exponential Functions as Mathematical Models is called an exponential function with base b and exponent x. The number of model F cameras that an employee with 1, 2, and 6 months of.

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