logarithmic functions just another way to write exponents
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Logarithmic FunctionsJust 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<a<1 and increasing if a > 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
Solve Change equation to
exponential equation
3
2
log (4 7) 2
4 7 3
4 7 9
4 16
4
x
x
x
x
x
Logarithmic Equation
Solve 2
log 64 2
64
8
log .
8
x
x
x
Domain of function is positive numbers
x
Logarithmic Equation
Solve: 2 5
ln ( )
ln 5 2
ln50.805
2
xe
Take on both sides of equation inverse of e
x
x
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 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.
0.350
nA A e
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