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(iv) If 0 < x1 < x2 then ln x1 < ln x2 We already know that the logarithm satis es the multiplicative property By applying this property repeatedly, we obtain that: If x > 0 and n is any integer then ln(x n ) = n ln x A companion result is the division rule: If a and b are positive numbers then a = ln a ln b ln b EXAMPLE 61

Meaning Converts v into a long integer and returns the result Converts v into a 1-byte array and returns the result Converts v into a 2-byte array and returns the result Converts v into an 8-byte array and returns the result

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Converts v into a 4-byte array and returns the result Converts v into a 4-byte array and returns the result Converts v into an 8-byte array and returns the result Converts v into a 2-byte array and returns the result Converts v into a 4-byte array and returns the result Converts v into an 8-byte array and returns the result Converts v into a 2-byte array and returns the result Converts v into a double value and returns the result Converts the byte at a[idx] into its bool equivalent and returns the result A non-zero value is converted to true; zero is converted to false Converts two bytes starting at a[start] into its char equivalent and returns the result Converts eight bytes starting at a[start] into its double equivalent and returns the result Converts two bytes starting at a[start] into its short equivalent and returns the result Converts four bytes starting at a[start] into its int equivalent and returns the result Converts eight bytes starting at a[start] into its long equivalent and returns the result Converts four bytes starting at a[start] into its float equivalent and returns the result

Practical Application of Rational B-Splines The rational B-spline method used to interpolate n motion constraints can be expressed as follows:

public static char ToChar(byte[ ] a, int start) public static double ToDouble(byte[ ] a, int start) public static short ToInt16(byte[ ] a, int start) public static int ToInt32(byte[ ] a, int start) public static long ToInt64(byte[ ] a, int start) public static float ToSingle(byte[ ] a, int start)

Part II:

SOLUTION We can write A in simpler terms by using the multiplicative and quotient properties: A = ln(a 3 b2 ) ln(c 4 d) = [ln a 3 + ln(b2 )] [ln(c 4 ) + ln d] = [3 ln a + 2 ln b] [( 4) ln c + ln d] = 3 ln a + 2 ln b + 4 ln c ln d The last basic property of the logarithm is the reciprocal law: For any x > 0 we have ln(1/x) = ln x EXAMPLE 62

S( x ) = A j R j , k ( x )

Method public static string ToString(byte[ ] a)

Meaning Converts the bytes in a into a string The string contains the hexadecimal values associated with the bytes, separated by hyphens Converts the bytes in a, beginning at a[start], into a string The string contains the hexadecimal values associated with the bytes, separated by hyphens Converts the bytes in a, beginning at a[start] and running for count bytes, into a string The string contains the hexadecimal values associated with the bytes, separated by hyphens Converts two bytes starting at a[start] into its ushort equivalent and returns the result Converts four bytes starting at a[start] into its uint equivalent and returns the result Converts eight bytes starting at a[start] into its ulong equivalent and returns the result

j =1

public static string ToString(byte[ ] a, int start)

SOLUTION We calculate that ln(1/7) = ln 7, ln(9/5) = ln 9 ln 5 = ln 32 ln 5 = 2 ln 3 ln 5 You Try It: Simplify ln(a 2 b 3 /c5 )

public static string ToString(byte[ ] a, int start, int count)

public static ushort ToUInt16(byte[ ] a, int start) public static uint ToUInt32(byte[ ] a, int start) public static ulong ToUInt64(byte[ ] a, int start)

(511)

TABLE 21-13 Methods De ned by BitConverter (continued)

Now you will see why our new de nition of logarithm is so convenient If we want to differentiate the logarithm function we can apply the Fundamental Theorem of Calculus: d d ln x = dx dx More generally, d 1 du ln u = dx u dx EXAMPLE 63

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