Exponential & Logarithmic equations By: Carina Amaya So are there differences between the Exponential and Logarithmic functions? Yes in fact their are, logarithmic and exponential functions are inverses of each other. • The domain of the exponential function is a set of real numbers, but the domain of the logarithmic function is a set of positive real numbers. The range of the exponential function is a set of positive real numbers, but the range of the logarithmic function is a set of real numbers. Math Rules ! Algebra II Semester 1 Project Two types of Exponential equations What is needed in an exponential equation? Exponential Equations Differences ?? How useful are these functions? An exponential function is a function whose value is a constant raised to the power of the argument. The function where the constant is E. Exponential functions can be used in many ways in real life such as modeling population growth, bacterial decay, compute investments and savings, even help coroners determine time of death. Logarithmic Equations Who discovered Logarithms? A mathematician by the name of John Napier, published a book in 1614 on his theory of logarithms. The purpose of his discovery was to make multiplication of larger quantities (sines then called at the time) easier. 2 log x = 12

We want to isolate the log x, so we divide both sides by 2.

log x = 6

Since log is the logarithm base 10, we apply the exponential function base 10 to both sides of the equation.

10log x = 106

By logarithmic identity 2, the left hand side simplifies to x.

x = 106 = 1000000 Example Exponential functions

used in real life. By: Carina Amaya Logarithmic function

applied to real life An exponential equation in which each side can be expressed in

terms of the same base can be solved using the property:

If the bases are the same, set the exponents equal. Exponential Growth

Exponential Decay Four variables are needed in an exponential equation percent change, time, amount at the beginning of that period, and the amount at the end of that period. When an original amount increases at a constant rate over a period of time that is Exponential Growth When an original amount decreases at a constant rate over a period of time that is Exponential Decay Growth:

Cell Phone Users In 1985, there were 285 cell phone subscribers in the small town of Centerville. The number of subscribers increased by 75% per year after 1985. How many cell phone subscribers were in Centerville in 1994? (Don't consider a fractional part of a person.)

Years x = (1)1986 (2)1987 (3)1988 (4)1989 (5)1990 (6)1991 (7)1992 (8)1993 (9)1994

Number of

Cell Phone

users : 498 872 1527 2672 4677 8186 14325 25069 43871

There are 43871 subscribers in 1994.

Function: y= a(1+r)*

a = the initial amount before the growth begins

r = growth rate

X = the number of intervals

y= 285(1+.75)* as X ranges from 1 to 9 for this problem

horizontal axis = year (1986 = 1)

vertical axis = number of cell phone users Examples Decay by half-life:

The pesticide DDT was widely used in the United States until its ban in 1972. DDT is toxic to a wide range of animals and aquatic life, and is suspected to cause cancer in humans. The half-life of DDT can be 15 or more years. Half-life is the amount of time it takes for half of the amount of a substance to decay. Scientists and environmentalists worry about such substances because these hazardous materials continue to be dangerous for many years after their disposal.

For this example, we will set the half-life of the pesticide DDT to be 15 years.

Let's mathematically examine the half-life of 100 grams of DDT.

End of Half life cycle Grams of DDT remaining:

1 15 yrs 50

2 30 yrs 25

3 45 yrs 12.5

4 60 yrs 6.25

5 75 yrs 3.125

6 90 yrs 1.5625

7 105 yrs .78125

8 120 yrs .390625

9 135 yrs .1953125

10 150 yrs .0976562

By looking at the pattern, we see that this decay can be represented as a function:

Function:

a = the initial amount before the decay begins

r = decay rate

x = the number of intervals

as x ranges from 1 to 10 for this problem horizontal axis = end of half life cycle

vertical axis = grams of DDT remaining Describing numbers in terms of 10 are

logarithms. You can see logarithmic functions being used all over such as in interest rates or rate of returns, and can be used in the Richter scale to measure the magnitude of earthquakes. The decibel system also uses logarithms when measuring sound and intensity. How can these be used to predict things? Exponential functions can be used to predict future growth and decay over a certain amount of time. Logarithms can be used to predict how intense something will be, or how much money you might have overtime normally using a previous pattern. New Text Message ! Hey that's me!

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# Exponential & Logarithmic and Life

Algebra II S1 Project

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