Chapter 3: Exponentials and Logarithms

Chapter 3: Exponentials and Logarithms

1. Logarithms

   • Solve
     • One-to-One Property
       • when a logarithmic equation has the same base on each side, the arguments must be equal (i.e log(2)=log(x-3)
     • Power Property
       • the log of a power is equal to the power times the log of the base
     • Quotient Property
       • the log of a quotient is equal to the difference of the logs of the numerator and denominator
     • Product Property
       • the log of a product is equal to the sum of the log of the first base and the log of the second base
     • Change-Of-Base Property
       • the log of a new base is the log of the new base divided by the log of the old base in the new base
   • Evaluate
     • Graphically
     • Symbolically
     • Numerically
       • No-Calculator
   • Natural and Common Logs
     • Natural
       • A natural log is a log with a base of e
     • Common
       • A common Log is a log with a base of 10

2. Exponential

   • Solving
     • Analytical
       • Product Property
         • when multiplying two powers with the same base you, you can just add the exponents (i.e. a^2 *a ^4 = a^6)
       • Quotient Property
         • when you divide powers with the same base you can just subtract the exponents
       • One-to-One Property
         • when a logarithmic equation has the same base on each side, the arguments must be equal (i.e 2^4=2^x-3
       • Power Property
         • To find the power of a power you can just multiply the exponents
     • Graphing
       • you can solve by graphing by putting the left side of the equation into the calculator as one equation and putting the right side in as a separate equation. Find the value of the variable by calculating the intersection
   • Graphing
     • Transformations
       • The Parent function of an exponential is f(x)=b^x................................. When a graph is transformed, the equation is altered the following ways.................. For example, the function f(x)= 2(b)^(x-5)-8 is transformed in the following ways........... Right 5, Down 8, and is stretched b a factor of 2
   • Word Problems
     • Real-World Problems
       • Growth and Decay
         • N=Noe^kt
       • Interest Problems
         • Compound Interest
           • A=P(1+r/n)^nt
         • Continuous Interest
           • A=Pe^(rt)
       • Population
         • Compound Population
           • N=No(1+r)^kt
         • Continuous Population Growth
           • N=Noe^kt

3. Exponential functions and logarithmic function are inverses

   • Converting between Exponentials and Logarithms
     • A log, for example log2(8x)=4, can be converted to an exponential function by moving the base to the constant, the exponent to the exponent, and the number in the parentheses to the other side of the equals sign so that it looks like 2^4=8x

4. Both exponential and logarithmic functions are examples of Transcendental Functions because they cannot be solved with simple algebra

5. Find the best fit model

   • Use the linear regression model on your calculator to determine the best fit model