UNIT 2

UNIT 2

1. Chapter 4

   • 4.1 Logarithmic functions
     • Common log= Log base 10
     • if a is positive, then the log base a of x is... logaX=y then a^y= X
     • logaX=y because a^y=x
       • Example log10000=5 because 10^5=100000
     • Basic properties of logarithms
       • loga1=0
       • logaa=1
       • logaa^x=x
       • a^logax=x
   • 4.4 The Natural exponential and logarithmic functions
     • Natural exponential functions with base e
       • f(x)=e^x
     • natural logarithm function
       • logeX= lnx
     • compounded continuously
       • A(t)=Pe^rt
         • r= interest rate
         • p= amount invested
     • Exponential
       • f(t)=Ce^rt
         • C= initial value
         • r= instantaneous growth/decay rate
         • growth if r>0
         • decay if r<0
   • 4.2 Laws of logarithms
     • logaAB= LogaA+logaB
     • logaA/B= logaA-logaB
     • logaA^C=ClogaA
     • Change the base of a formula
       • logbx= logax/logab
   • 4.3 Logarithmic scales
     • Linear scale: each mark adds a fixed constant
     • Logarithmic scale: each mark multiplies by a fixed factor
     • ph scale measure acidity levels
       • ph= -log[H+]
         • H+ is the concentration of hydrogen ions in a solution
         • Example: ph=-log(10^-7) = -7
     • The decibel scale
       • S=klog(I/I0)
         • s= subjective sensitivity of the stimulus
         • I= Physical intensity
         • I0= Threshold physical intensity
         • k=constant
         • we use: B=10log(I/10^-12)
     • Richter scale
       • Magnitude (m) = log (I/S)
         • I= the earthquake intensity
         • s=sensitivety of a "standard" earthquake (M is 0)
       • An increase of t units on the R scale means is 10^t times as strong

2. Chapter 3

   • Toolkits: A.3 Integer exponents A.4 Radicals and exponents
     • a^n= axaxa... multiply a by itself n times
       • a= real number= base
       • n=positive integer= exponent
     • Rules for working with exponents
       • a^m x a^n = a^m+n
       • a^m/a^n = a^m-n
       • (a^m)^n = a^mn
       • (ab)^n = a^n b^n
       • (a/b)^n = a^n/b^n
       • a^0=1
       • a^-n = 1/a^n and 1/a^-n = a^n
     • Rational exponents
       • a^1/n = nsqrt{a}
         • When n is even then a is equal to or greater than
       • a^m/n= nsqrt{a^m}
   • 3.4 Graph of exponential functions
     • base a : f(x)=a^x
       • a>1 and a doesn't equal 0 ever
     • graphs of exponential functions
       • domain: all real numbers
       • Range: all positive real numbers
       • Growth
         • f(x)= a^x a>1
       • Decay
         • f(x) = a^x 0<a<1
   • 3.3 Comparing linear and exponential growth
     • Linear vs Exponential ( as the input increases by 1 unit)
       • Linear
         • constant average rate of change
         • y=mx+b
       • Exponential
         • constant percentage rate of change
         • y=Ca^x
   • 3.1 Exponential growth and decay
     • Exponential growth: f(x)=Ca^x
     • Growth
       • f(x)=Ca^x Where a > 1
       • x= The number of time periods
       • base a = the growth factor
       • C = Initial value
       • positive graph
     • Decay
       • f(x)=Ca^x Where 0<a<1
       • Decay rate (-#) : a=r+1
       • negative graph
     • growth rate r and growth factor a related by a=r+1
       • a>1
       • r>0
   • 3.2 Exponential models: Comparing rates
     • Changing the time period
       • changing from time period x to time period y, raise the original growth factor to the number of time periods x there are in the time period y
         • EXAMPLE: if the daily growth factor is a, then the weekly growth factor is a^7
     • Compound interest
       • A(t)= P(1+r/n)^ nt
         • n= times per year
         • P= invested amount
         • r= annual interest rate
         • t= years
         • A(t)= amount of the investment in the end
       • N: annually=1 monthly=12 quarterly=4 weekly=52 semi-annually=2 daily=365
     • Annual percentage yield (APY): annual growth rate of an investment
       • (1+r/n)^n1