1. Algorithms CSCI 235, Spring 2019 Lecture 5 Exponential and Logarithmic functions

2. Exponential functions
an = product of n copies of a
a-n = 1/an
Key identities:
a0 = 1
aman = am+n
(am)n = amn
Examples:
(22)3 = ?
253/2 = ?
2223 = ?

3. Logarithmic functions
logba = x, the power to which b must be raised to equal a
bx = a
Example: log28 = 3, because 23 = 8
Logarithm facts:
logb(1/a) = -logb(a)
logc(ab) = logc(a) + logc(b)
Special case: logc(an) = nlogc(a)
logc(1) = 0
clogc(b) = b = logc(cb)
logc(a) = logc(b) logb(a)

4. Notation for this course
lg(n) = log2(n)
ln(n) = loge(n) Natural log, e =
lgk(n) = (lg(n))k
Examples:
lg(2n3) = ?
ln(32) = ?
32(lg n) = ?

5. Comparing logs and exponentials
How do log2(n) and log3(n) compare?
How do 2n and 3n compare?

6. Useful Facts for comparing logs and exponentials
for real b, a > 1
What does this mean for the asymptotic relationship?
for real b, a > 0
What does this mean for the asymptotic relationship?

7. Factorial n! = 1*2*3* . . . *(n-1)*n Upper bound on n! is nn
in other words: n! = o(nn)
Stirling's approximation:
From this we can derive:
n! = w (2n)
n! = o(nn)
lg(n!) = Q(nlg n)

8. Summations
Basic equalities:

9. Arithmetic Series Definition of arithmetic series
where c is a constant
For arithmetic series:
Why?

10. Examples a) b)
=? ak = 3 + ak-1

11. Geometric Series Definition of geometric series where c is a constant
For geometric series:

12. Examples a) b) c)