Exponential function

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In mathematics, an exponential function is a function, a rule to find one number (y) from another number (x). The new number is found by multiplying a certain number, called the base (a), by itself a number of times which is equal to the first number. For example, y = 2^x is an exponential function, found by multiplying 2 by itself x times. If x = 3, then If x = 4, then A person can also multiply an exponential function by some number, and it will still be an exponential function. In general, an exponential function looks like this:

c can be any number and a can be any number bigger than zero. If a is between zero and one, then the exponential function becomes smaller as x becomes bigger. If a is bigger than one, then the exponential function becomes bigger as x becomes bigger. The exponential function is sometimes very big and sometimes very small, but it is always bigger than zero.

An important thing about exponential functions is that adding a number to x is the same as multiplying y by a number. For example, in the exponential function y = 2^x, adding 1 to x is the same as multiplying y by 2.

[change] Examples

The exponential function is the way that things grow, if they grow faster when they are bigger. For example, if one million people live in a country, then ten thousand babies are born every year, but if two million people live in that country, then twenty thousand babies are born every year. The way the number of people who live in the country becomes bigger every year is an exponential function.

Another example of something that becomes bigger and bigger in the same way as an exponential function is money in a bank. If a person puts 100 dollars in a bank, after one year the bank will give that person 110 dollars. The extra 10 dollars, or 10 percent, is called the interest. If the person puts the 110 dollars into the bank again, after another year the bank will give the person 121 dollars. The money the person has does not grow by the same amount every year, but it grows faster and faster. The bank does not give the person ten dollars after the first year and then ten dollars after the second year. The bank gives the person ten dollars after the first year and then eleven dollars after the second year. The way the amount of money in the bank becomes bigger every year is an exponential function.

An exponential function can say how things get bigger, but it can also say how they get smaller. For example, tritium is a radioactive kind of hydrogen. After some time, tritium changes into helium. If there are 1000 atoms of tritium in a bottle, 12 years later, only 50 atoms will be left. After another 12 years, only 25 will be left. The "half-life" of tritium is 12 years. The number of tritium atoms in the bottle is this exponential function:

t is the number of years after a person put 100 tritium atoms in the bottle.

[change] Relation to the mathematical constant e

Even though the base (a) can be any number bigger than zero, for example, 10 or 1/2, often it is a special number called e. The number e cannot be written exactly, but it is almost equal to 2.71828.

The number e is important to every exponential function. For example, a bank pays interest of 0.01 percent every day. One person takes his interest money and puts it in a box. After 10,000 days (about 30 years), he has 2 times as much money as he started with. Another person takes his interest money and puts it back into the bank. Because the bank now pays him interest on his interest, the amount of money is an exponential function. After 10,000 days, he doesn't have 2 times as much money as he started with, but he has 2.718145 times as much money as he started with. This number is very close to the number e. If the bank pays interest more often, so the the amount paid each time is less, then the number will be closer to the number e.

A person can also look at the picture to see why the number e is important for exponential functions. The picture has three different curves. The curve with the black points is an exponential function with a base a little smaller than e. The curve with the short black lines is an exponential function with a base a little bigger than e. The blue curve is an exponential function with a base exactly equal to e. The red line is a tangent to the blue curve. It touches the blue curve at one point without crossing it. A person can see that the red curve crosses the x-axis, the line that goes from left to right, at -1. This is true only for the blue curve. This is the reason that the exponential function with the base e is special.

e is the unique number a, such that the value of the derivative of the exponential function f (x) = a^x (blue curve) at the point x = 0 is exactly 1. For comparison, functions 2^x (dotted curve) and 4^x (dashed curve) are shown; they are not tangent to the line of slope 1 (red).