Expert: Paul Klarreich - 12/6/2011

Question
a.One important application of logarithms is found in various computer search routines. For example, a binary search algorithm on a table (or array) of data takes a maximum of log2n (“log base 2, of n”) steps to complete, where n is the number of data elements that can be searched. How many steps (at most) are needed for a search of a table with 16 elements? 512 elements? Explain.
b.The approximation of the natural logarithm of 2: ln 2 ≈ 0.693 is commonly used by applied scientists, biologists, chemists, and computer scientists. For example, chemists use it to compute the half-life of decaying substances. Based on this approximation and the power rule for logarithmic expressions, how could you approximate ln 8, without a calculator? Explain

Answer
Questioner: Patti
Country: Texas, United States
Category: Calculus
Private: No
Subject: Logarithms
Question: a.One important application of logarithms is found in various computer search
routines. For example, a binary search algorithm on a table (or array) of data takes a
maximum of log2n (“log base 2, of n”)

>>> Write  log2(n) -- always use parentheses here.


steps to complete, where n is the number of data elements that can be searched. How many
steps (at most) are needed for a search of a table with 16 elements? 512 elements?
Explain.

Now  X = log2(n)  means   2^X = n.

So if the number of elements, n = 16,

write   X = log2(16),  means  2^X = 16.

And if the number of elements, n = 512,

write   X = log2(512),  means  2^X = 512.

Just work out your powers of 2 now.


b.The approximation of the natural logarithm of 2: ln 2 ≈ 0.693 is commonly used by
applied scientists, biologists, chemists, and computer scientists. For example, chemists
use it to compute the half-life of decaying substances. Based on this approximation and
the power rule for logarithmic expressions, how could you approximate ln 8, without a
calculator? Explain

Use this property of the logarithm function:

ln(x^n) = n ln(x)

Now if ln(2) is known, and we know 2^3 = 8,  write  ln(8) = ln(2^3) = 3 ln 2.

You can finish up.