Logarithms

















What are the values of the following Logarithms?


	log10(1000)

	log2(16)



What's the Logarithm?
What's the Base of the Logarithm?


	if the log[b] of n is k, then b^k = n

















What's an easy way to approximate the log base 10 of n?


	round n to the closest power of 10 and count the zeros



What's an easy way to approximate the log base 2 of n?


	count the number of bits needed to represent n in binary

















Classwork
You may work with a partner.

Give the approximate value of the log base 10 for each number.

	123
	9999
	72457
	2183746058


Give the approximate value of the log base 2 for each number.

	1000
	1000*1000
	1000*1000*1000

	123
	9999
	72457
	2183746058

















An important truth about logs.

	log(a*b) = log(a) + log(b)


How does log(n) compare to n for large values of n?


How does log2(n) compare to log10(n) for various values of n?
How does the number of bits in a binary number compare to
the number of digits in an equivalent decimal number?

	log2(n) is about 3.3 * log10(n)


When logarithms are used in Big-Oh formulas,
why is the base of the logarithm not given?

















Log's in Big-Oh's


If you start with X = 1, how many times you can double X
before X becomes greater than or equal to N?

If you start with X = N, how many times you can cut X in half
before X becomes less than or equal to 1?


When you are analyzing code to find the Big-Oh,
when do you use a logarithm in the Big-Oh formula?

















Classwork
You may work together with a partner.

Give the Big-Oh of the running time for each program fragment.


	for ( int x = 1; x < n; x *= 2 )
	    sum++;

	for ( int i = 0; i < n; i++ )
	    for ( int x = n; x > 1; x /= 2 )
	        sum++;

	for ( int x = n; x > 1; x /= 10 )
	    sum++;

	for ( int x = 1; x < n; x *= 10 )
	    for ( int i = 0; i < n; i++ )
	        sum++;

















Classwork
You may work together with a partner.


If an algorithm is O(log n) and it takes 1 second to solve a problem
of size 10, how long will it take to solve a problem of size 1000?


If an algorithm is O(n log n) and it takes 1 second to solve a problem
of size 10, how long will it take to solve a problem of size 1000?


If an algorithm is O(log n) and it takes 10 seconds to solve a problem
of size 100, how large a problem can be solved in 1000 seconds?

















Where does O(log n) fall between O(1) and O(n)?

Is it halfway between?
Is it closer to O(1)?
Is it closer to O(n)?



Where does O(n log n) fall between O(n) and O(n^2)?

Is it halfway between?
Is it closer to O(n)?
Is it closer to O(n^2)?