Exponential & Logarithms

Exponential & Logarithms

1. Log Properties

   • Product Property
     • The logarithm of a product is the sum of the logarithms.
       • Log(b)mn=Log(b)m+Log(b)n
   • Division Property
     • The logarithm of division is the subtraction of the logarithms.
       • Log(b)m/n=Log(b)m-Log(b)n
   • Power Property
     • The exponent in a logarithm is moved to the front of the logarithm.
       • Log(b)m^n=nLog(b)m
   • Change of Base Formula
     • To get a base of 10 in a Logarithm, you have to divide the Log of the answer by the Log of the base.
       • Log(b)X=LogX/Log(b)

2. Solving

   • Logarithm Equations
     • log(4)x=log(4)3+log(4)(x-2)
       • *condense into single logarithms on each side*
         • log(4)x=log(4)(3x-6)
           • *if the bases are the same, the logarithms will cancel each other out*
             • x=3x-6
               • *solve for x*
                 • x-x+6=3x-x-6+6 6/2=2x/2 3=x
   • Exponential Equations
     • 4^(x+2)=16^(x-3)
       • *find the smallest number that each coefficient can be squared by*
         • (2^2)^(x+2)=(2^4)^(x-3)
           • *distribute the exponents*
             • 2^(2x+4)=2^(4x-12)
               • *set the exponents equal to each when the bases are the same*
                 • 2x+4=4x-12
                   • 2x-2x+4+12=4x-2x-12+12 16/2=2x/2 8=x

3. Graphing

   • Solve Exponential Equations
   • Logarithms w/o Calculator
   • Transformations
     • Expoential
       • Graph the parent function.
         • Create a table of values for the exponential function f(x)=b^x
           • ex: y=2+3^x-1
             • -1: Horizontal Shift right 1 2: Vertical Shift up 2
     • Logarithms
       • Graph the parent function.
         • Create a table of values for the logarithmic function f(x)=log(b)x
           • ex: y=-log(5)(x-1)+3
             • -1: Horizontal Shift right 1 -log: Reflection over X-axis 3: Vertical Shift up 3

4. Word Problems

   • Money
     • Compound Interest
       • A=P(1+r/n)^nt
     • Continuously Compounded Interest
       • A=Pe^rt
   • Population
     • Exponential Growth
       • N=N(o)(1+r)^t
     • Continuous Exponential Growth
       • N=N(o)e^kt

5. The log function is the inverse of the exponential function and vice versa.

   • Converting
     • Equivalent Exponential and Logarithmic Form
       • In logarithmic notation: e=log(b)n
         • e- exponent b- base n- answer
       • In exponential form: n=b^e

6. Real-World Problems

   • Make a scatter plot of the data given.
     • Plug data points into calculator table.
       • Analyse the graph with the data points, determining if the points are increasing/decreasing.
         • If the points are increasing, then you use the exponential equation. If the points are decreasing, you use the logarithm equation.

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