Logarithmic Functions Quiz Notes

Logarithmic Functions
  • Definitions
    • A logarithmic function is the inverse of an exponential function.
    • The general form of a logarithmic function is:
      f(x)=extlogb(x)f(x) = ext{log}_b(x)
      where ( b ) is the base of the logarithm.
Key Equations
  • Logarithmic to Exponential: A logarithmic equation can be converted to an exponential form:
    extify=extlogb(x), then x=byext{if } y = ext{log}_b(x) \text{, then } x = b^y
  • Domain and Range:
    • The domain of a logarithmic function is ( (0, +\infty) ) since logarithms are undefined for non-positive values.
    • The range is ( (-\infty, +\infty) ).
Inverse Functions
  • To find the inverse function of a logarithmic function:
    • If ( f(x) = ext{log}_b(x) ), then the inverse function ( f^{-1}(x) = b^x ).
    • For example, the inverse of ( f(x) = ext{log}_7(x) ) is ( f^{-1}(x) = 7^x ).
Examples

  • Example of calculations:

    • Logarithmic values:
    1. extlog20(20)=1ext{log}_{20}(20) = 1
    2. extln(0.5)≈−0.693ext{ln}(0.5) \approx -0.693
    3. extlog(0.25)≈−0.602ext{log}(0.25) \approx -0.602

  • Exponential Decay:
    The exponential function modeling radioactive decay:
    a(t)=50e−0.04ta(t) = 50e^{-0.04t}
    Determines time to reduce from 50 grams to 25 grams:

    • Set ( a(t) = 25 ):
      25=50e−0.04t  ⟹  0.5=e−0.04t25 = 50e^{-0.04t} \implies 0.5 = e^{-0.04t}
      extTakingnaturallogarithm:ln⁡(0.5)=−0.04t  ⟹  t=−ln⁡(0.5)0.04ext{Taking natural logarithm: } \ln(0.5) = -0.04t \implies t = \frac{-\ln(0.5)}{0.04}
Graphing Logarithmic Functions
  • Transformation Properties:
    • Vertical stretches, translations, and reflections can be applied to logarithmic functions:
    • Example:
      • Stretching ( f(x) = ext{log}(x) ) by a factor of 3 results in ( g(x) = 3 ext{log}(x) ).
    • To apply a transformation:
    1. Vertical shift by ( k ) means ( f(x) + k ).
    2. Horizontal shift by ( h ) means ( f(x-h) ).
Understanding Population Growth
  • Exponential Model:
    • Given population data:
      | Time (hours) | Population |
      |--------------|------------|
      | 0 | 1000 |
      | 1 | 1500 |
      | 2 | 2250 |
      | 3 | 3375 |
    • Exponential growth model:
      P(t)=1000(1.5)tP(t) = 1000(1.5)^t
Logarithmic Regression
  • Perform logarithmic regression using a graphing calculator to create a logarithmic model for time as a function of population. Check for accuracy by validating against actual data values.
Miscellaneous
  • pH Calculation:
    • pH is calculated using the formula:
      pH=−log⁡[H+]pH = -\log[H^+]
      where ( [H^+] ) is the concentration of hydrogen ions in moles per liter. For example, for ( [H^+] = 1.5 \times 10^{-9} ), the calculation is:
      pH=−log⁡(1.5×10−9)pH = -\log(1.5 \times 10^{-9})