Researach websites on exponential and logarithmic

Exponential and Logarithmic Functions

Name XXXXX

School XXXX

Professor XXXXX

Subject XXX

Exponential and Logarithmic Functions

Introduction

This presents briefly the definitions , examples , characteristics and applications of both exponential and logarithmic functions

Exponential Functions ? 1 , and x , the exponent , is any real number (Marcus , 2008 Roberts 2008a Stapel , 2008a . Figure 1 presents three examples of exponential graphs of the forms f (x 2x , f (x 8x , and f (x 0 .5x (Roberts 2008a

The characteristics of the three graphs are summarized as follows (1 the domain

of f (x ) is all real numbers (2 ) the range of f (x ) is all positive real numbers (3 ) when b 1 , the graph and its slope increase when x increases (4 ) when 0 b 0 (6 ) the graph is asymptotic to the x-axis - it does not cross the x-axis or touch it , but it gets closer and closer to the x-axis as x gets smaller and smaller (Roberts , 2008a Stapel , 2008a

Exponential functions are useful in real world situations . Common applications include analyzing population growth , computing compound interest , carbon dating artifacts , determining time of death , and calculating exponential decay . To show an application using an exponential function , consider the credit cards with interest compounded daily . The function describing the balance (principal interest ) on an average daily balance of 500 at interest rate of 15 will be : C (t 500 (1 .15 /365 )365t , where t represents the number of days for calculating the interest (Taylor , 2008

Logarithmic Functions ? 1 , and x , the exponent , is any positive real number . The function f (x logb (x ) is the inverse of the exponential function f (x bx , or y logb (x ) is equivalent to x b y (Roberts , 2008b Stapel , 2008b Figure 2 presents three examples of logarithmic graphs of the forms y log (x , y log (x /log (2 , and y log (x /log .5 (Roberts , 2008b

The characteristics of the three graphs are summarized as follows (1 the domain of f (x ) is all positive real numbers (2 ) the range of f (x is all real numbers (3 ) when b 1 , the graph increases when x increases (4 ) when 0 b 0 (6 ) the graph is asymptotic to the y-axis - it does not cross the y-axis or touch it but it gets closer and closer to the y-axis as y gets smaller and smaller (Roberts , 2008b Stapel , 2008b

Logarithmic functions are likewise useful in real world situations , such as in estimating the number of species , determining the magnitude of earthquakes or the intensity of sound waves , determining the acidity of a solution , and calculating the rate of producing fresh water from salt water (Saye , 2008 . To show an application in using logarithmic function , consider a solution 's pH is defined by

(t -log10 (t where t is the hydronium ion concentration in moles per liter . The function for finding the pH of a solution with hydronium ion concentration 4 .5 x 10-5 will be

(t -log10 (4 .5 x 10-5 (The...