[size=150][color=#ff0000]NEEDING TO BRUSH UP THE EXPONENTS[/color][/size][br]We have seen that if a quantity [math]\large{A}[/math] changes following an exponential behaviour with respect of a variable [math]\large{x}[/math], it can expressed by something like:[br][br][math]\Large{A=A(x)=A_0 \cdot f^x}[/math][br][br]where [math]\large{A_0}[/math] is the starting amount of [math]\large{A}[/math] (when [math]\large{x=0}[/math]) and [math]\large{f}[/math] if the constant factor by which [math]\large{A}[/math] changes.[br][br]We define [b]exponential function[/b] any function where the independent variable appears as exponent, such as[br] [br][math]\large{y=2^x}[/math][br][br]Before trying to handle these objects, we need to understand what it means having a power whose exponent can assume [b]any[/b] real number. To do this, let's review the meaning of some particular powers we already know.[br][br][size=150][color=#ff0000]REVIEWING POWERS WITH "SPECIAL" EXPONENTS[br][/color][/size]We all know what a power is; for example [math]\Large a^4[/math] means[br][math]\Large a^4=\underbrace{a \cdot a \cdot a \cdot a}_{4\ times}[/math][br]Now let's see what happens when using some unusual exponents.[br] [br][size=100][color=#0000ff][b]ZERO EXPONENT[/b][/color][/size][br]Using what we know about [b][color=#ff0000]the properties of powers[/color][/b] we can obtain that [math]\Large a^0=1[/math] (if [math]\Large a[/math] is NOT zero).[br][br]As a matter of fact we have that [math]\Large a^0[/math] [b]can be considered the result of a division like[/b] [math]\Large a^3 : a^3 = a^{3-3} = a^0[/math], and this tells us that:[br][br][math]\Large a^0=a^{3-3}=a^3:a^3=\frac{a \cdot a \cdot a}{a \cdot a \cdot a} = 1[/math][br][br]where in the last step we simplified all elements both in the numerator and in the denominator: as there is the same number of elements in the numerator and in the denominator, in both parts of the fractions we cancel out everything and we are left with 1.[br][br][size=100][color=#0000ff][b]NEGATIVE EXPONENT (exponent is an [i]integer[/i] number)[/b][/color][/size][br]Reasoning in a similar way we can obtain that [math]\Large a^{-3}=\frac{1}{a^3}[/math].[br][br][b]Once more we use the property of division with like bases[/b], and we consider [math]\Large a^{-3}[/math] as the result of a division like [math]\Large a^2 : a^5 = a^{2-5} = a^{-3}[/math]. Considering the calculations we have that:[br][br][math]\Large a^{-3}=a^{2-5}=a^2:a^5=\frac{a \cdot a}{a \cdot a \cdot a \cdot a \cdot a} = \frac{1}{a \cdot a\cdot a} = \frac{1}{a^3}[/math][br][br]When simplifying the elements, we have three elements left in the denominator, because 5 exceeds 2 by 3.[br][br]This works exactly in the same way of[br][br][math]\Large a^5:a^2=\frac{a \cdot a \cdot a \cdot a \cdot a}{a \cdot a} = a \cdot a \cdot a = a^3[/math][br][br]In this case the number of elements in the dividend exceeds the one in the divisor, so there are some elements left in the [i]numerator[/i] (i.e. the [i]upper[/i] side of the fraction). When the exponent is negative we simply have that there are more elements in the divisor, and therefore some elements remain in the [i]denominator[/i], i.e. in the [i]lower[/i] side of the fraction.[br][br]Please note that in general a negative exponent is equivalent of taking the reciprocal of the base, so[br][math]\Large \left(\frac{2}{3}\right)^{-3} = \left(\frac{3}{2}\right)^{3}[/math][br][br][size=100][color=#0000ff][b]FRACTIONARY EXPONENT (exponent is a [i]rational[/i] number)[/b][/color][/size][br]Now we will prove that [math]\Large a^{\frac{1}{2}}=\sqrt{a}[/math].[br][br]To obtain that, [b]we must remember the definition of "[i]square root of [math]a[/math][/i]"[/b]: this is the number which, raised to the second, gives us [math]\Large a[/math]. If the raise to the second [math]\Large a^{\frac{1}{2}}[/math] we get:[br][br][math]\Large \textcolor{red}{(}a^{\frac{1}{2}}\textcolor{red}{)^2}=a^{\frac{1}{2} \cdot 2} = a^1 = a[/math][br][br]This means that [math]\Large a^{\frac{1}{2}}[/math][b] behaves exactly as the square root of [math]a[/math], and therefore they are the same number[/b]. Obviously this works for [i]any[/i] fractionary exponent, so we have that[br][br][math]\Large a^{\frac{1}{3}} = \sqrt[3]{a}[/math] as [math]\Large \textcolor{red}{(}a^{\frac{1}{3}}\textcolor{red}{)^3}=a^{\frac{1}{3} \cdot 3} = a^1 = a[/math][br][br]Please note that [math]\Large a^{\frac{2}{3}}[/math] is the cube root of [math]\Large a^2[/math], as [br][br][math]\Large a^{\frac{2}{3}} = \sqrt[3]{a^2}[/math] as [math]\Large \textcolor{red}{(}a^{\frac{2}{3}}\textcolor{red}{)^3}=a^{\frac{2}{3} \cdot 3} = a^2 [/math][br][br]In general we have that[br][br][math]\Large a^{\frac{n}{m}} = \sqrt[m]{a^n}[/math][br][br]The denominator of the exponent is the index of the root, where the numerator is the usual exponent.[br][br][size=150][color=#ff7700][b]IRRATIONAL EXPONENT (exponent is a [i]real[/i] number): ???[/b][/color][/size][br]As we want to consider [math]\Large a^x[/math] with [math]\Large x[/math] being [i]any[/i] number, we must ask ourselves what does an [i]irrational[/i] exponent means.[br][br]For example, what's the meaning of [math]\Large a^{\sqrt{2}}[/math]?[br][br]The answer is not easy, so let's help ourselves with a practical application of exponential functions. We will do it studying the behaviour of waterlily, the nice flower you can see below.[br][br][img width=230,height=172]https://gardendrama.files.wordpress.com/2012/02/attraction.jpg[/img][br][br]In the animation below we will discover that the behaviour of this plant, as supposed in the problem, can be described by an exponential function. We will draw the function giving to the exponent the values we have studied until now. After the animation we will try to go further introducing irrational exponents.

[color=#ff0000][size=150]WHAT NEXT?[/size][/color][br]In the example of waterlilies we introduced an exponential function evaluating it for: [br][list][*][b]natural exponents[/b], i.e. after a given number of weeks[/*][*][b]negative exponents[/b], which proved to reproduce the behaviour of n weeks [i]ago[/i][/*][*][b]fractionary (rational) numbers[/b], which proved to fit into the curve when evaluating the amount after a non integer number of weeks. [br][/*][/list][br][b]Yet, we haven't evaluated our function in all points were [/b][math]x[/math][b] (or [/b][math]w[/math][b], in our example) is an irrational number[/b]. For instance what will the extensions of waterlilies be after [math]\sqrt{2}[/math] weeks?[br][br]The question sounds a little bit weird, and indeed it is. [br][br]Nonetheless we will try to find an answer in next animation.

[size=150][color=#ff0000]A BRIEF PREVIEW OF LIMIT CONCEPT[/color][br]Let's summarize what we have seen:[br][list][*][b]any irrational number con be approximated as precisely as we want with a rational quantity[/b]. For instance [math]\large{\sqrt{2}\approx 1,4142}[/math][br][br][/*][*][b]as long as we have a rational number, we know how to consider it as an exponent[/b], so we get an approximation for the value we are looking for. For instance [math]\Large{2^{\sqrt{2}}\approx 2^{1,4142}=2^\frac{14142}{10000}=\sqrt[10000]{2^{14142}}}[/math] (ok, you probably may find it not very nice, but we do know how to calculate it!)[br][br][/*][*][b]to get a better approximation, we simply need to consider more decimals[br][br][/b][/*][*]so [b]we can "virtually" define [math]\large{2^{\sqrt{2}}}[/math] as the number obtained considering as many decimals of [math]\large{\sqrt{2}}[/math] as we need[/b]. [/*][/list]We will see later on during our studies that this kind of approach is called [i]a limit[/i]: we defined the value of a power with an irrational exponent as the number we obtain going [i]nearer and nearer[/i] to that irrational value (without reaching it actually, as we do not know how to calculate it). The trick is that we get nearer to it (we [i]approach it[/i], as it is said using technical terms) [b]considering rational exponents, whose power we can calculate with no difficulty[/b]. [br][br]We can represent this process of [i]approaching[/i] the irrational value using the symbol of [i]limits, [/i]which we will meet much later when studying analysis:[br][br][math]\huge \lim_{x_n\to \sqrt{2}}a^{x_n}=a^{\sqrt{2}}[/math][br][br]The preceding expression is formally expressed as follows:[br][br]"[color=#0000ff]the limit of[/color] [math]\Large \textcolor{blue}{a^{x_n}}[/math], [color=#ff0000]as[/color] [math]\Large \textcolor{red}{x_n}[/math] [color=#ff0000]approaches[/color] [math]\textcolor{red}{\sqrt{2}}[/math], [color=#0000ff]is equal to[/color] [math]\Large \textcolor{blue}{a^{\sqrt{2}}}[/math]"[br][br], which can be translated in more common words as: [br][br]"[color=#ff0000]the closer [the approximation of] [/color] [math]\Large \textcolor{red}{x_n}[/math] [color=#ff0000]gets to [/color][math]\textcolor{red}{\sqrt{2}}[/math],[color=#0000ff] the closer[/color] [math]\Large \textcolor{blue}{a^{x_n}}[/math] [color=#0000ff]gets to[/color] [math]\Large \textcolor{blue}{a^{\sqrt{2}}}[/math]"[br][br]NOTE: we used the symbol [math]\Large x_n[/math], and not simply [math]\Large x[/math], as it is intended that [math]\Large x[/math] is approaching [math]\sqrt{2}[/math] passing [i]by a discrete set of values[/i], that is not all possible values but only [i]some[/i] of them (which we can call [math]\Large x_1,\ x_2,\ x_3...[/math]) which are its rational approximations: [math]\Large 1.4[/math] then [math]\Large 1.41[/math], then [math]\Large 1.1414, 1.14142, 1.141421, 1.1414213, \hdots[/math] [br][br]We can use the same symbols describe the behaviour of the function in the left side of the graph: the greater and negative it gets the [math]\Large x[/math] (this is expressed saying that [math]\Large x[/math] [i]tends to a negative infinite quantity[/i], represented by [math]\Large -\infty[/math]), the more the power approaches to [math]0[/math].[br][br][math]\huge \lim_{x \to - \infty} 2^x = 0[/math][br][br]This is read "[i]the limit of [math]\Large 2^x[/math] when [math]\Large x[/math] approaches negative infinity is zero[/i]"[br][br][color=#ff0000]GENERAL FEATURES OF EXPONENTIAL FUNCTION [/color][math]\Large y=a^x[/math][/size][br]Let's now see the general features common to all exponential function like the one we studied in the case of waterlilies.[br][color=#0000ff][br]BASE [math]\Large \textcolor{blue}{a}[/math] [u]MUST[/u] BE GREATER THAN ZERO[/color][br]In our example the base of the power was [math]\Large 2[/math], but obviously we could have taken another number as base. The first important thing we must take into account, however, is that [b]WE CANNOT CONSIDER AN EXPONENTIAL WITH NEGATIVE OR ZERO BASE, therefore it must be [math]\Large a>0[/math][/b].[br][br]As a matter of fact, as the exponent [math]\large x[/math] wil range between [i]any[/i] real number, [b]the exponent will assume also values like [math]\large \frac{1}{2}, \frac{3}{4}, \frac{7}{8}[/math], etc..., which are interpreted as roots with even index[/b]. We know that such roots must have non negative radicand, so if we want to evaluate an exponential function for [i]any[/i] value of [math]x[/math] as exponent, we must exclude functions with negative bases.[br][br]We exclude also [math]\large a=0[/math], because [math]\large y=0^x[/math] would be a rather boring functions resulting always [math]\large 0[/math] (and giving us many troubles when [math]\large x=0[/math]), so we are not interested in studying it.[br][br]This limitation will be present also with logarithms, which are powers seen from a different point of view.[br][br][color=#0000ff]EXPONENTIAL FUNCTION NEVER GETS NEGATIVE[/color][br] As a direct consequence of the requirement described in previous paragraph, we have that [math]\large a^x[/math] is always a positive number, and therefore the exponential function [math]\large y=a^x[/math] NEVER gets on or below the [math]\large x[/math] axis. You can see this in next image.

[color=#0000ff]BEHAVIOUR CHANGES IF [/color][math]a>1[/math][color=#0000ff] OR [/color][math]a<1[/math][br]In the waterlilies example we condidered and exponential function with [math]a=2[/math]. Its behaviour is similar when [math]a[/math] assumes many different values, [b]but it is not always the same for any value of [/b][math]a[/math]. We see this feature in the next animation.

A functions like the exponential with base greater than [math]1[/math] is called [i][color=#38761d][b]increasing function[/b][/color][/i] ("funzione monotona crescente", in Italian), because in [i]any[/i] of its parts you have that the greater gets the [math]x[/math], the more increases the [math]y[/math] (i.e the function's result). Another example of increasing function is a line with positive angular coefficient.[br][br]A functions like the exponential with base less than [math]1[/math] is called [i][color=#ff0000][b]decreasing function[/b][/color][/i] ("funzione monotona decrescente", in Italian), because in [i]any[/i] of its parts you have that the greater gets the [math]x[/math], the more decreases the [math]y[/math] (i.e the function's result). Another example of increasing function is a line with negative angular coefficient.[br][br]Finally please note that for the exponential function taking the reciprocal of the base has the same effect of changing the sign of the exponent, that is: the same effect of evaluating the function for [math]-x[/math] instead of [math]x[/math]. For this reason two exponential functions having reciprocal bases are [b]symmetrical[/b] with respect of the [math]y[/math] axis: considering the first one for a given [math]x_1[/math] point gives the same result as considering the other one for [math]-x_1[/math] (that is the symmetrical of [math]x_1[/math] with respect of the y axis). You can verify this feature drawing point by point two such functions, like the ones used in previous animation or [math]y=3^x[/math] and [math]y=\left(\frac{1}{3}\right)^x[/math].