**6.1 Introduction: **The functions y = xn are power functions, so polynomials are made from power functions. More general exponents arise naturally in applications. For example, volume increases as the (3/2)th power of the surface area. "Root" functions such as the square root of x and cube root of x are power functions.

Power functions are functions of the form f(x) = x2, f(x) = 4 x-3, f(x) = -9 x1/2, and so on. The general form is f(x)= kx^{p} where p is any real number and k is non-zero.

**6.2 Domains of power function:** If p is a non-zero integer, then the domain of the power function f(x) = kxp consists of all real numbers. For rational exponents p, xp is always defined for positive x, but we cannot extract an even root of a negative number. Thus x^{(1/4)} is not defined for any negative real numbers. Neither is x^{(3/4)} (the fourth root of x cubed).

Any rational number p can be written in the form p = r/s where all common factors of r and s have been cancelled. When this has been done, kx^{p} has domain

- All real numbers if s is odd
- All non-negative real numbers if s is even.

If p is a real number which is not rational (called an irrational number), then the domain of x^{p} consists of all non-negative real numbers

**6.3 Logarithm as power function**: Logarithms give us away around this as the **Power Function** is in fact an **Antilogarithm**. The math behind writing a power function, using the above rules, is as follows:

ln(*x*^{y}) = y x ln(*x*)

*x*^{y} = e^{y x ln(x)}

**Some basic log rules**:

- i) log (
*x*x y) = log (*x*) + log (y) - ii) log(
*x*/y) = log(*x*) – log(y) - iii) log(
*x*^{a}) = a x log(*x*) - iv) log
_{z}(*x*) = log_{y}(*x*)/logy(z)

Logarithms only exist for Positive values, i.e. values that are strictly greater than zero. 0 raised to any power is defined as 0, and negative numbers raised to powers are only defined for powers of integers.

**6.4 Exponents as power function:**

The Rules of exponentiation are:

- i)
*x*^{(p + q)}= (*x*^{p})(*x*^{q}). - ii)
*x*^{pq}= (*x*^{p})^{q}. - iii)
*x*^{-p}= 1/*x*^{p}. - iv)
*x*^{(1/p)}is the p-th root of*x*. - v)
*x*^{0}= 1 for any*x*0. 0^{0}is undefined. - vi) (
*x*y)^{p}= (*x*^{p})(y^{p}).

Domain: all numbers.

Range: all positive numbers.

If a (the base > 1, exponent is an increasing function, it is less than 1 for all negative *x*, and above 1 for all positive x. If a < 1, it is a decreasing function, greater than one for *x* < 0, and less than 1 for *x* greater than or equal to 0.

**6.5 Calculation**

The equation involving a power function is an equation of the form

k*x*^{p} = b.

The solution of such an equation, if a solution exists, is given by

*x* = (b/k)^{(1/p)}.

and it can be easily checked using the rules for exponentiation.

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