Is it possible to use pi as the base of an exponential function

The two letters l and n are reversed from the order in English because it arises from the French logarithm naturalle. It is used much more frequently in physics, chemistry, and higher mathematics than other logarithmic functions. The natural logarithm function can be used to solve equations in which the variable is in an exponent. Privacy Policy. Skip to main content. Exponents, Logarithms, and Inverse Functions. Search for:.

The Real Number e. Math books and even my beloved Wikipedia describe e using obtuse jargon:. The natural logarithm, formerly known as the hyperbolic logarithm, is the logarithm to the base e, where e is an irrational constant approximately equal to 2.

Nice circular reference there. No more! Save your rigorous math book for another time. Pi is the ratio between circumference and diameter shared by all circles.

It is a fundamental ratio inherent in all circles and therefore impacts any calculation of circumference, area, volume, and surface area for circles, spheres, cylinders, and so on. Pi is important and shows all circles are related, not to mention the trigonometric functions derived from circles sin, cos, tan. Just like every number can be considered a scaled version of 1 the base unit , every circle can be considered a scaled version of the unit circle radius 1 , and every rate of growth can be considered a scaled version of e unit growth, perfectly compounded.

So e is not an obscure, seemingly random number. Let's start by looking at a basic system that doubles after an amount of time. For example,. Splitting in two or doubling is a very common progression. Sure, we can triple or quadruple, but doubling is convenient, so hang with me here. As a general formula:.

Clever, eh? So the general formula for x periods of return is:. Our formula assumes growth happens in discrete steps. Our bacteria are waiting, waiting, and then boom , they double at the very last minute. Our interest earnings magically appear at the 1 year mark.

Based on the formula above, growth is punctuated and happens instantly. The green dots suddenly appear. If we zoom in, we see that our bacterial friends split over time:.

After 1 unit of time 24 hours in our case , Mr. Green is complete. He then becomes a mature blue cell and can create new green cells of his own. The equation still holds. But money is different. As soon as we earn a penny of interest, that penny can start earning micro-pennies of its own. So, we earn 50 cents the first 6 months and another 50 cents in the last half of the year:. Sure, our original dollar Mr. Blue earns a dollar over the course of a year.

But after 6 months we had a cent piece, ready to go, that we neglected! That 50 cents could have earned money on its own:. Who says we have to wait for 6 months before we start getting interest? Make sense? But see that each dollar creates little helpers, who in turn create helpers, and so on.

Why not take even shorter time periods? How about every month, day, hour, or even nanosecond? Will our returns skyrocket? Our return gets better, but only to a point. Try using different numbers of n in our magic formula to see our total return:. This limit appears to converge, and there are proofs to that effect. But as you can see, as we take finer time periods the total return stays around 2. The number e 2. But with each tiny step forward you create a little dividend that starts growing on its own.

When all is said and done, you end up with e 2. Aside: Be careful about separating the increase from the final result. What can we do here? In other words, the function tends to the constant value when x tends to minus infinity.

This concept helps to find the asymptotes of exponential functions, which is shown as below:. The curve represents the general form of an exponential function. As x becomes smaller and smaller, the curve tends to become a straight line.

This line is called the asymptote of the exponential function. Asymptotes of exponential functions are always horizontal lines and hence it can be concluded that an exponential function has only one horizontal asymptote. If the value of b is 0, then x-axis is the asymptote of the exponential function. If it is negative, then the asymptote will be below and parallel to the x-axis. Note that in this case the base is 4 and the exponent is x - 1.

Step 1: Find ordered pairs. I have found that the best way to do this is to do it the same each time. Step 2: Plot the points and draw the curve. Example 2: The population of bacteria in a culture is growing exponentially. At there were 80 bacteria present and by PM there were bacteria.

You need to find the values of the parameters k and a. Example Since 3 is a constant, we will concentrate on the second part of the equation to begin with, namely 0. Rewriting 0.