Chapter 3

Chapter 3

1. Exponential Expression

   • A real number that is multiplied by  a positive integer (or sometimes a negative integer) 
   • 10^6
     • 10 to the 6th power 

2. Exponential Growth & Decay

   • When the base is multiplied and increases by a time period 
   • f(x)=Ca^x
     • x is the number of time periods ; c is the initial value
   • Growth : a> 1
     • I.e. f(x)= 45(1.03) ^x
       • Growth adds any number to 1, so the growth or rate factor is over 1. 
     • growth factor
       • Growth factor is a; which is the factor that f(x) increases when x is multiples by a time period. 
   • Decay: a=1+r
     • I.e. f(x) =25(.95)^x
       • For decay, the decay factor will bless than 1. 
     • Decay Factor
   • Changing Time Periods
     • Later/ Initial
       • This is used to find the percent difference of two values to create an exponential growth or decay equation. 
     • Half-Life
       • The time it takes for half the atoms to decay. It is used to describe data in a much quicker way. 
       • Half-Lifes will always be 1/2 or .5
     • Compounding Interest
       • A=P (1 + r/n)^nt
         • *A is the amount in the account  *P is the amount initially deposited  *r is the annual interest (decimal)  *n is the number of times the interest is compounded  *t is how long the investment is made for 
     • Compounded Continuously
       • A=Pe^rt
         • e is the irrational number represented by the letter.  e is about 2.718281828
       • Annual Yield Percentage
         • (1 + r/n)^nt -1

3. Exponential Decay

4. Chapter 4 : Logs

   • "log base a of x"
     • logs are inverses of exponents . Exponential equations can be written as logs and vice versa. 
     • Change of Base Formula
       • log b (x)= log a (x) / log a (b)
         • I.e. log (355) /log (11)
   • Laws of Logs
     • log b (mn)=log b (m) + log b (n)
       • When multiplying two logs you have to expand the two and add them together 
     • log b (m/n) = log b (m)- log b (n)
       • When dividing logs, you have to subtract the logs from one another. 
     • log b (m^n) = n * log b (m)
       • the log of a power is the exponent multiplied by the log base. 
   • Log Scales
     • pH scale
       • Measures how acidic or basic a substance is 
       • pH=log(H^+)
         • H is the concentration of hydrogen ions 
     • Decibal Scale
       • Measurement of sound 
       • S= k* log (I/I 0)
     • Richter Scale
       • Used to calculate magnitude of earthquakes 
       • M= log (I/S)
   • Natural Exponential & Log Functions
     • Natural Expo. : f(x) =e^x
     • Natural Log : f(x) =In (x)= log e (x)