Logarithmic Functions Just another way to write exponents.

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Presentation transcript:

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 Best Tutorial that I’ve found

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 1.

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.

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

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

Logarithmic Equations SolveChange equation to exponential equation

Logarithmic Equation Solve

Logarithmic Equation Solve:

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

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%?

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).

Healing of Wounds The normal healing of wounds can be modeled by an exponential function. If the formuladescribes 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.

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?

Additional Examples More Examples of Logarithmic Equations