Transcript 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
 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
log 3 (4 x  7)  2
4 x  7  32
4x  7  9
4 x  16
x4
Change equation to
exponential equation
Logarithmic Equation

Solve log x 64  2
x 2  64
x  8
Domain of log function is positive numbers.
x 8
Logarithmic Equation

Solve: e2 x  5
Take ln on both sides of equation (inverse of e)
ln 5  2 x
ln 5
x
 0.805
2
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%?
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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 A  A e 0.35 n describes the area of
0
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?
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Additional Examples

More Examples of Logarithmic Equations