Fall 2015 COMP 2300 Discrete Structures for Computation Donghyun (David) Kim Department of Mathematics and Physics North Carolina Central University 1 Chapter 11.4 Exponential and Logarithmic Functions: Graphs and Orders

Graphs of Exponential Functions The exponential function with base b > 0 is the function that sends each real number x to. 2 Fall 2015 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University x

Graphs of Exponential Functions – cont’ Relationship between exponential functions and polynomial function [Wiki][Wiki] 3 Fall 2015 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University

Graphs of Logarithmic Functions If b is a positive real number not equal to 1, then the logarithmic function with base b,, is the function that sends each positive real number x to the number, which is the exponent to which b must be raised to obtain x. If b > 1, then for all positive number and, if, then. 4 Fall 2015 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University

Based 2 Logarithms of Numbers between Two Consecutive Powers of 2 Prove the following property Q) If k is an integer and x is a real number with, then. 5 Fall 2015 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University

Number of Bits in a Binary Representation How many binary digits are need to represent following? (3 bits = ) (5 bits = ) In general, k binary digits are sufficient to represent numbers How many binary digits are needed to represent 52,837 in binary notation? – 16 binary digits 6 Fall 2015 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University k bits

Exponential and Logarithmic Order For all real numbers b and r with b > 1 and r > 0, for all sufficiently large real numbers x, and for all sufficiently large real numbers x. For all real numbers b and r with b > 1 and r > 0, is, and is. For all real numbers b with b < 1 and for all sufficiently large real numbers x, For all real numbers b > 1, 7 Fall 2015 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University

Order of a Harmonic Sum Sums of the form are called harmonic sums. Show that 8 Fall 2015 COMP 2300 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University