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Logarithmic Functions

Logarithmic Functions. Just another way to write exponents. Definition, Domain & Graphs. Definition Relating Logarithms to Exponents Changing Exponential Expressions to Logarithmic Expressions and vice versa Finding the Exact Value of a Logarithmic Expression Domain Graphs

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Logarithmic Functions

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  1. Logarithmic Functions Just another way to write exponents

  2. Definition, Domain & Graphs • Definition • Relating Logarithms to Exponents • Changing Exponential Expressions to Logarithmic Expressions and vice versa • Finding the Exact Value of a Logarithmic Expression • Domain • Graphs • Best Tutorial that I’ve found

  3. Properties of a Logarithmic Function • The domain is the set of positive real numbers; the range is all real numbers (inverse of exponential function) • The x-intercept of the graph is 1. There is no y-intercept • The y-axis (x = 0) is a vertical asymptote of the graph • A logarithmic function is decreasing if 0<a<1 and increasing if a > 1.

  4. Properties of a Logarithmic Function • The graph of f contains the points (1, 0), • (a,1), and 1/a, -1). • The graph is smooth and continuous, with no corners or gaps.

  5. Natural Log Function • Definition • Y = ln x if and only if x = e^x

  6. Common Logarithm Function • Definition • Y = log x if and only if x = 10^y • Graphs using transformations

  7. Logarithmic Equations • Solve Change equation to exponential equation

  8. Logarithmic Equation • Solve

  9. Logarithmic Equation • Solve:

  10. Alcohol and Driving • The concentration of alcohol in a person’s blood is measurable. Recent medical research suggests that the risk R (given as a percent) of having an accident while driving a car can be modeled by the equation • R = 6e^kt • where x is the variable concentration of alcohol in the blood and k is a constant

  11. Alcohol and Driving • Suppose that a concentration of alcohol in the blood of 0.04 results in a 10% risk (R = 10) of an accident. Find the constant k in the equation. Graph the equation using this k. • Using this value of k, what is the risk if the concentration is 0.17? • Using the same value of k, what concentration of alcohol corresponds to a risk of 100%?

  12. Alcohol and Driving • If the law asserts that anyone with a risk of having an accident of 20% or more should not have driving privileges, at what concentration of alcohol in the blood should a driver be arrested and charged with a DUI (Driving Under the Influence).

  13. Healing of Wounds • The normal healing of wounds can be modeled by an exponential function. If the formula describes the area of a wound on the nth day following an injury when no infection is present to retard the healing. Suppose that a wound initially had an area of 100 sq. mms.

  14. Healing of Wounds • (a) If healing is taking place, how many days should pass before the wound is one-half its original size? • (b) How long before the wound is 10% of its original size?

  15. Additional Examples • More Examples of Logarithmic Equations

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