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.

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