]> Exponent and Log

Chapter 1: Differentiation; Section 2: Real Functions; page 4

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Exponent and Log

Exponential function

The definition of an exponential function is:

 $y={a}^{x}\text{ }\text{for}\left\{\begin{array}{l}a>0\\ a\ne 1\end{array}\right\}$ (1.2.4.1)

The constant  a  is called its base and it should be positive for a domain of the argument (x) covering all the real numbers. The requirement of  $a\ne 1$  excludes the trivial case of  y=1 . By definition the exponential function is positive at any point of its domain, it is increasing for base  a > 1  and decreasing for  a < 1 .

For reasons which will be clarified in the next section, it is common to use as the base, an irrational number called the natural base, defined by:

 $e=\underset{n\to \infty }{\mathrm{lim}}{\left(1+\frac{1}{n}\right)}^{n}=2.71828....$ (1.2.4.2)

The exponential function with the natural base is called the exponent, and will be denoted by

 $y=\mathrm{exp}x={e}^{x}$ (1.2.4.3)

The use of the notation  expx  is preferable than  ${e}^{x}$  in view of a possible ambiguity e.g. in the case of  ${e}^{\frac{1}{2}}=±\sqrt{e}$ , while the exponent by definition is non-negative. For this reason we'll use exclusively the notation  expx .

The exponent is plotted and its values are shown for three points of  x  in Fig. Exponent.

We already know that a quantity bearing physical dimensions can be raised to a power. The number expressing the power itself is by definition a pure number, therefore dimensionless. It means that the variable  x  of the exponential function (1.2.4.1) should be dimensionless. On the other hand the base  a  cannot have a physical dimension either, since if the function  y  is a physical entity, it cannot have a variable dimension.

An example for use of the exponent function in physics is the law of radioactive decay:

 $N={N}_{0}\mathrm{exp}\left(-\frac{t}{T}\right)$ (1.2.4.4)

where  N  is the amount of radioactive material left after a time  t ,  ${N}_{0}$  is the amount of radioactive material at the beginning (t = 0) and  T  is a time constant called the lifetime. For  t = T  the amount of radioactive material decreases by a factor of  e  (N = ${N}_{0}$/e) .

However an apparent deviation of this rule may occur for the use of physical dimensions with the exponential function. For instance a closer look at (1.2.4.4) shows that it is perfectly legal to rewrite

according to the rule  ${a}^{\left(p\text{ }q\right)}={\left({a}^{p}\right)}^{q}$ , and therefore the two powers bearing dimensions appear, but they are connected to the same base and actually the dimensions cancel each other.

Logarithmic function

The inverse of the exponential function with base  a  (1.2.4.1) is called the logarithmic function (with the same base  a) and is written as:

 $y={\mathrm{log}}_{a}x\text{ }\text{for}\text{ }\left\{\begin{array}{l}a>0\\ a\ne 1\end{array}\right\}$ (1.2.4.5)

In other words the function expresses the power on which the base  a  should be raised in order to obtain the argument (x). For example if the base is  a = 10 , then the logarithm yields the order of magnitude.

From the definition and the properties of the exponential function, it follows that the domain of the logarithmic function consists of all the positive numbers and that the function is increasing for  a > 1  and decreasing for  a < 1 .

Since the logarithmic and the exponential functions are each one inverse of the other, it follows:

 ${a}^{{\mathrm{log}}_{a}x}={\mathrm{log}}_{a}\left({a}^{x}\right)=x$ (1.2.4.6)

The logarithm has important properties when applied to a product and to a power.

Logarithm of a product:

 ${\mathrm{log}}_{a}\left(b\text{ }c\right)={\mathrm{log}}_{a}b+{\mathrm{log}}_{a}c$ (1.2.4.7)
Proof: from after applying  ${\mathrm{log}}_{a}$  on the first and last terms and using (1.2.4.6) one obtains (1.2.4.7).

Logarithm of a power:

 ${\mathrm{log}}_{a}\left({b}^{c}\right)=c\text{ }{\mathrm{log}}_{a}b$ (1.2.4.8)
Proof: from after applying  ${\mathrm{log}}_{a}$  on the first and last terms and using (1.2.4.6) one obtains (1.2.4.8).

One can change the base of the logarithm with:

 ${\mathrm{log}}_{a}x={\mathrm{log}}_{a}b\text{\hspace{0.28em}}{\mathrm{log}}_{b}x$ (1.2.4.9)

This can be obtained from (1.2.4.8) by substituting  c  by  ${\mathrm{log}}_{b}x$ .

As consequence of (1.2.4.9), by substituting  x  by  a  one obtains the relation:

 ${\mathrm{log}}_{b}a\text{\hspace{0.28em}}{\mathrm{log}}_{a}b=1$ (1.2.4.10)

As in the case of the exponential function, it is common to use the natural base also for logarithm. The logarithm using the natural base is called natural logarithm (lan) and we'll adopt the common notation:  lnx . Since it is the inverse function of  expx  we can rewrite (1.2.4.6) for the natural base:

 $\mathrm{exp}\left(\mathrm{ln}x\right)=\mathrm{ln}\left(\mathrm{exp}x\right)=x$ (1.2.4.11)

The graphical display of  lnx  is obtained by inversion of the  expx  at Fig. Lan .

The logarithm to any base can be obtained from the natural logarithm by simple scaling:

 ${\mathrm{log}}_{a}x=\frac{\mathrm{ln}x}{\mathrm{ln}a}$ (1.2.4.12)

that follows from (1.2.4.9) and (1.2.4.10).

The exponential function to any base can be obtained from  expx  by simple scaling of the argument:

 ${a}^{x}=\mathrm{exp}\left(x\text{ }\mathrm{ln}a\right)$ (1.2.4.13)

This is illustrated in Fig. Exponential Function. The proof of (1.2.4.13) is left as a simple exercise for the user.

Since the logarithmic function is the inverse of the exponential function, the same rule about physical dimensions take place, namely both the argument and the function cannot have physical dimensions.

Here also, as in the case of the exponential function, there is an apparent inconsistency with the rule about the use of physical dimensions. As an example let's rewrite (1.2.4.4) by applying lan on both sides. One obtains

including two lan functions with arguments bearing dimensions, but they appear in a subtraction of two lans, which is equivalent to one lan with an argument consisting of a dimensionless ratio.

Hyperbolic functions

The parity of a function was introduced at page 2 of the present section. The exponent does not have a parity, but as shown at (1.2.2.3) can be written as a sum of even and odd parity functions. The even function is called hyperbolic-cosine:

 $\mathrm{cosh}x=\frac{\mathrm{exp}x+\mathrm{exp}\left(-x\right)}{2}$ (1.2.4.14)

and the odd - hyperbolic-sine:

 $\mathrm{sinh}x=\frac{\mathrm{exp}x-\mathrm{exp}\left(-x\right)}{2}$ (1.2.4.15)

The similarity of the names with the trigonometric functions is not accidental, but has a meaning as it will be clarified when the complex numbers will be used. By continuing this similarity, another odd function is defined, namely the hyperbolic-tangent:

 $\mathrm{tanh}x=\frac{\mathrm{sinh}x}{\mathrm{cosh}x}$ (1.2.4.16)

From the definition one obtains:

 $\underset{x\to ±\infty }{\mathrm{lim}}\left(\mathrm{tanh}x\right)=\underset{x\to ±\infty }{\mathrm{lim}}\frac{\mathrm{exp}x-\mathrm{exp}\left(-x\right)}{\mathrm{exp}x+\mathrm{exp}\left(-x\right)}=±1$ (1.2.4.17)

In order to complete the list of the hyperbolic functions, here are the seldom used reciprocals as in the trigonometric case:

 $\begin{array}{l}\text{sech}\text{\hspace{0.17em}}x=\frac{1}{\mathrm{cosh}\text{\hspace{0.17em}}x}\\ \text{csch}\text{\hspace{0.17em}}x=\frac{1}{\text{sinh}\text{\hspace{0.17em}}x}\\ \text{coth}\text{\hspace{0.17em}}x=\frac{1}{\mathrm{tanh}\text{\hspace{0.17em}}x}\end{array}\right\}$ (1.2.4.18)

From the definitions (1.2.4.14) and (1.2.4.15) it is left for the user to deduce the relation:

 ${\mathrm{cosh}}^{2}x-{\mathrm{sinh}}^{2}x=1$ (1.2.4.19)

More properties of the hyperbolic functions are similar (but not identical) to those of the trigonometric functions. The following relations are few example:

 $\begin{array}{l}\mathrm{sinh}\left(a±b\right)=\mathrm{sinh}a\text{\hspace{0.17em}}\mathrm{cosh}b±\mathrm{cosh}a\text{\hspace{0.17em}}\mathrm{sinh}b\\ \mathrm{cosh}\left(a±b\right)=\mathrm{cosh}a\text{\hspace{0.17em}}\mathrm{cosh}b±\mathrm{sinh}a\text{\hspace{0.17em}}\mathrm{sinh}b\\ \mathrm{tanh}\left(a±b\right)=\frac{\mathrm{tanh}a±\mathrm{tanh}b}{1±\mathrm{tanh}a\text{\hspace{0.17em}}\mathrm{tanh}b}\end{array}\right\}$ (1.2.4.20)

where  a  and  b  are any two numbers. These relations follow from the definition of the hyperbolic functions. The exercise 1 deals with their proof (see below).

The hyperbolic functions are obtained and plotted at Fig. Hyperbolic functions

The important properties of the hyperbolic functions are summarized here:

• sinhx is odd, increasing and unbounded.
• coshx is even, increasing for positive x and its minimum is equal to +1 .
• tanhx is odd, increasing and bounded between −1 and +1.

Inverse hyperbolic functions

As in the case of the inverse trigonometric functions, the inverse hyperbolic functions also bear the prefix "arc" and are written with an "a" in front.

The properties of the arc-hyperbolic functions follow from the properties of the hyperbolic functions. At this stage you have already the necessary knowledge to figure it out:

For any  coshx  value they are two possible values of the argument, therefore the  acoshx  is double valued, with one branch of negative values and another with - positive. Its domain is  $x\ge 1$ .

asinhx  is odd, increasing and unbounded in all the domain of real numbers as cosequence of the  sinhx  properties.

Since  tanhx  is bounded between  −1  and  +1 , the  atanhx  is defined only for the interval of the argument (−1, +1) . It is odd, increasing and

 $\underset{x\to ±1}{\mathrm{lim}}\left(\text{atanh}\text{\hspace{0.17em}}x\right)=±\infty$ (1.2.4.21)

The relations of the arc-hyperbolic functions to  lan  follow:

 $\begin{array}{l}\text{asinh}\text{ }x=\mathrm{ln}\left(x+\sqrt{{x}^{2}+1}\right)\text{ }\text{for all}\text{\hspace{0.28em}}\text{\hspace{0.28em}}x\\ \text{acosh}\text{ }x=\mathrm{ln}\left(x±\sqrt{{x}^{2}-1}\right)\text{ }\text{for}\text{\hspace{0.28em}}x\ge 1\\ \text{atanh}\text{ }x=\frac{1}{2}\mathrm{ln}\left(\frac{1+x}{1-x}\right)\text{ }\text{ }\text{\hspace{0.28em}}\text{for}\text{\hspace{0.28em}}|x|<1\end{array}\right\}$ (1.2.4.22)

The exercise 2 deals with the derivation of these relations (see below).

The graphic presentation of the arc-hyperbolic functions together with the corresponding hyperbolic functions are given as an interactive do-it-yourself presentation in Fig. Hyperbolic functions and inverse.

Exercises

Exercise 1. Prove the equations (1.2.4.20) by means of the definitions (1.2.4.14-16).

Exercise 2. Derive the relations (1.2.4.22) by inverting the hypebolic functions as defined by (1.2.4.14-16)!

Why does the square-root in the expression of asinhx not have a double sign?
Pay attention to derive and not to prove (1.2.4.22)!

Exercise 3. In the following expression the physical dimensions of x and y are denoted by [x] and [y] accordingly:

What are the physical dimensions of the constants: a, b and c ?

Exercise 4. Discuss the properties of the following functions in terms of: the domain, points of discontinuity (inside the domain), intervals of increase and of decrease, and bounds.

1. $y=\mathrm{ln}\left(-x\right)$
2. $y=\mathrm{ln}|x|$
3. $y=\mathrm{ln}\left(\mathrm{sin}x\right)$
4. $y=\text{atanh}\left(\mathrm{tan}x\right)$
5. $y=\text{atan}\left(\mathrm{tanh}x\right)$

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