1. log(ab) = log a + log b 2. log(a)= clog a 3. log a log b log, a 4. blog, a = a <= Answer the following:: (a) Using induction and fact 1, prove fact 2 for positive integers c.
1. log(ab) = log a + log b 2. log(a)= clog a 3. log a log b log, a 4. blog, a = a <= Answer the following:: (a) Using induction and fact 1, prove fact 2 for positive integers c.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:Facts:
1. log(ab) = log a + log b
2. log(a) = clog a
log a
3.
log b
4. blog, a = a
= log, a
=
Question 5
Answer the following::
(a) Using induction and fact 1, prove fact 2 for positive integers c.
(b) Use fact 1 and fact 2 to show that log = loga - log b.
(c) It is often said that the bases of logarithms do not matter when finding asymptotic bounds, although
this idea is sometimes used too freely. To understand when we can ignore base, use fact 3 to show
that logan = = (log, n) for any positive a, b.
=
(d) A common mistake some students make is claiming that clog, n = O(n). Sometimes, they claim this
function is exponential. Give a counterexample for both claims (find a positive c and b in which
clog, n is neither exponential in n nor linear in n).
(e) Complete the argument above by showing that for all positive c and b, clog, n
=
(n) for some
k (this implies that such functions are never exponential in n, but the base does matter!). Your
argument should use facts 2 and 4.
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