Math Assignment Help With Exponents And Logarithm As Power Function

Chapter 6 Exponents And Logarithm As Power Function

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:

1. i) log (x x y) = log (x) + log (y)
2. ii) log(x/y) = log(x) – log(y)
3. iii) log(x^a) = a x log(x)
4. 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) x⁰ = 1 for any x0. 0⁰ is undefined.
• vi) (xy)^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

kx^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.

Email Based Assignment Help in Exponents And Logarithm As Power Function

To submit Exponents And Logarithm As Power Function assignment click here.