Homework Help is Here – Start Your Trial Now!
Math
Advanced Math
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.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter1: Fundamental Concepts Of Algebra
Section1.2: Exponents And Radicals
Problem 92E
See similar textbooks
icon
Related questions
Question

Vipul 

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.
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.
Expert Solution
trending now

Trending now

This is a popular solution!

steps

Step by step

Solved in 3 steps with 2 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Recommended textbooks for you
Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
SEE MORE TEXTBOOKS