Algebraic Theory of Exponents and Radicals

Algebraic (Exponential and Logarithmic) Theory
of powers and roots

This page derives the theory of radicals and exponents from algebraically described properties of the natural logarithms and exponential function. Student may assume calculator buttons provide the values of the latter.

A full development of theory or natural logarithms and the exponential appears in calculus books. This site provides its version of that development in chapter 19 of Volume 3, Why Slopes & More Math. Chapter 19 is included in webpages 19 Logs & Powers 19 Natural Logarithms. 19 Exponential Function. Optional readings, challenging perhaps, for now and later.

Hint: Assume the properties properties of the natural logarithms and exponential function first and see their derivation later.

Natural logarithm and exponential functions with the aid of an electronic calculators

Calculator button exercises met above suggest relationships between x² and x³ buttons, the a^x or y^x functions, and how to obtain a^x = exp(x ln(a)) via a sequence of operations on the calculator. The latter requires a > 0 but allows x to be any real number. In the case of the natural e, ln(e) = 1 and e^x = exp(x ln(e)) = exp(x). So we can use the e^x button on the calculator to compute exp(x). The latter function should not be confused with the EXP button present on some calculators.

A. Natural Logarithms:

The following properties can be illustrated or confirmed with an electronic calculator.

ln(x) is defined only when x > 0 and not defined when x < 0.
Fundamental Properties of Logarithms:

ln(ab) = ln(a) + ln(b)

when a > 0 and b > 0.
ln(1) = 0 from fundamental property or from a calculator.
ln (1/a) = - ln(a)
ln(aⁿ ) = n ln(a) for each whole number - demonstrate with numbers and even proofs for small values of n.
ln(1/aⁿ ) = -n ln(a) for each whole number n as ln(1/b) = ln

B. Exponential Function

The existence of the exponential function exp(x) can be assumed, and then computed with the aid of the e^x button on a calculator.

Alternatively, the points on the graph of y = ln(x) can be plotted and interpolated. Then exp(c) can be defined as the value of the x-coordinate of the intersection of the horizontal line y = c with the graph y = ln(x). The latter gives an example of the horizontal line method for calculating a function or defining an inverse function. The use of this horizontal line method to calculate exp(c) and the use of the vertical line method to calculate ln(d) gives a pair of operations or functions, each of which reverses the other. The formal definition of iverse function can be skipped or included here. .

Main Properties

The following properties can be derived from the properties of the natural logarithm and the horizontal line method for defining exp(x) from the graph of y = ln(x). Alternatively some or all properties can be assumed without proof, and the rest, if any, derived from the assumed ones. In either case numerical examples are needed to empirically illustrate all the following properties.

Domain: exp(x) = e^x is defined for all real numbers x, at least those we can enter into the display of an electronic calculator
Fundamental Property of Exponentials:

exp(a+b) = exp(a) * exp(b).
exp(0) = 1
exp(x) exp(-x) = 1 or exp(-x) = 1 / exp(x)
ln(exp(x)) = x when x is real - so the natural log function ln reverses the calculation of the exponential function exp (x),
exp(ln(x)) = x when x > 0 - so exponential function exp reverses the calculation of the natural log function ln(x),

C. Powers with whole number exponents

If we put f_a(n) = exp(n ln(a)) then deriving the following

f_a(1) = a^¹
f_a(2) = a^²
f_a(3) = a^³.
f_a(4) = a^⁴.

or demonstrating these patterns numerically suggests the general pattern f(n) = aⁿ or

aⁿ = exp(n ln(a))

for whole numbers n. The property exp(-x) = 1/exp(x) then implies

1
aⁿ

exp(-n ln(a))

D. Powers with real number exponents

Saying how to compute a number, defines it.

Suppose a is a positive number and x is a real number. Then compute the power a^x by the formula

a^x = exp(x ln(a))

In the power a^x the number a > 0 is called the base and x is called the exponent.

Two properties a^x a^y = a^x+y and (a^x)^y = a^xy can be also suggested and verified in numerical exercises. Details of how these algebraically described properties follow from the algebraically described properties of the exponential function and natural logarithms follow.

Derivation of a^x+y= a^x a^y

a^x+y= exp((x+y) ln(a)) by definition of a^x+y

= exp(x ln(a) + y ln(a)) by distributive law

= exp (x ln (a)) exp(y ln(a))

due to fundamental property:

exp( c + d) = exp (c) exp(d)

for all real numbers c and d.

= a^x a^yby definition of a^x and a^x

Derivation of (a^x)^y = a^xy

(a^x)^y = exp( y ln (a^x))

   = exp( y ln ( exp [ x ln(a ) ] )

   = exp( y [x ln(a)] )
   = exp( [yx] ln(a) )
   = exp ( [xy] ln(a) ) = a^xy

E. m-th Roots (m whole) for positive real numbers

Suppose a is a positive number, and m is a whole number.

Assume a = 64 and m = 2 on first reading. Then assume a = 64 and m = 3 on second reading.

We will try to solve the equation

y^m = a

for y where y is a positive number.

Solution: The natural logarithm of both sides of the equation y^m = amust be equal since y^m and a are two different ways of writing or representing the same number on paper. Therefore

ln(y^m ) = ln(a)

But the left side ln(y^m ) = m ln (y) due to the fundamental property ln(pq) = ln(p) + ln (q) of natural logarithms when p and q are positive. Therefore

m ^. ln(y) = ln(a)

The latter implies

and so

That is

should satisfy

y^m = a

The m-th root of a is

F. Even Roots of Real Numbers

Saying how to compute a quantity defines it.

Recall a positive times a positive is positive, zero times zerio is zero and negative times a negative is positive. Therefore x² > 0 for all real numbers x.

Square Roots

Because x² > 0 for all real numbers x, the equation

x² = a

only has solutions x when a > 0, that is only when a is non-negative. Defining

_
Öa

as the nonnegative real solution of x² = a works only if a is positive or zero. For a = 0,

_
Öa = 0

For a > 0, the solution is provided by the computation

See above. The latter is called the principal or positive square root. The negative square root is obtained by taking the negative of the principal square root. It is given by

All Further Even Roots

Similarly, if n = 2m > 0 is an even whole number, then xⁿ = x^2m= x^mx^m > 0 for all real numbers x. Thus the equation

x^2m = a

only has a solution x when a > 0, that is only when a is non-negative, since x^2m= x^mx^m > 0 for all real numbers x. The solution x = 0 when a = 0.

For a > 0, the positive solution, called the principal root, is

There is also a negative solution

G. Odd Roots of Real Numbers

Saying how to compute a quantity defines it.

The identify x = sign(x) |x| and consequences

Each real number x = sign(x) |x|. For instance

+5 = (+1) 5 as sign (5) = +1 and |+5| = 5 = distance of +5 = 5 to origin 0
-4 = (-1) 4 as sign (4) = -1 and |-4| = 4 = distance of -4 to origin 0
0 = (0)(0) as sign(0) = 0 and |0| = 0 = distance of 0 to itself.

Now sign(x) = +1, 0 or -1. In all, three cases [sign(x)]²= 1. Therefore

x³ = [sign(x)]|x|³

In general,

x^2m+1 = [sign(x)]|x|^2m+1

since (+1)^2m+1 = +1 and (-1)^2m+1 = -1.

Cube Roots

For a in nonzero, the equation x³ = a implies or requires

|y|³ = |a|

Therefore

When a is positive, taking y = |y| implies

y³ = |a| = a.

But when a is negative, taking y = (-1)|y| gives

y³ =(-1)³ |y|³ = - |a| = a.

The foregoing is equivalent to saying

satisfies y³ = a whenever a is nonzero. For a non-zero, that is a > 0 or a < 0, let

and let

All further Odd Roots:

For a in nonzero, and n a whole number, the equation x²ⁿ⁺¹ = a implies or requires

|y|²ⁿ⁺¹ = |a|

Therefore

When a is positive, taking y = |y| implies

y²ⁿ⁺¹ = |a| = a.