November 13, 2013
*Extra Problem (3.2)
Find the logistic function given the following:
limit to growth = 30, initial value = 5, passes through (3,15)
November 13, 2013
3.3 Logarithmic Functions
y=bx
Does the function: y = bx have an
inverse?
What does it look like?
domain:
range:
The inverse of an exponential function f(x)=bx is a
logarithmic function with base b
y = logb(x) iff
by = x
November 13, 2013
Given an equ. in exp. form, write it in log form.
Exp. Form
1. 23=8
2. 102=100
3. 4-2=
1
16
4. 34=81
Log. Form
November 13, 2013
Properties:
logb1 = 0
because b0 = 1
logbb = 1
because b1 = b
logbby = y
because by = by
b
logbx
=x
because logbx = logbx
November 13, 2013
Evaluate.
1. log2128
2. log.2125
3. log3x=4
4. logx 1 =-4
81
5. log127 = x
3
November 13, 2013
common logs
base 10
natural logs
base e
y = log x iff
y = ln x
y
10 = x
iff ey =x
November 13, 2013
Evaluate.
1. log 100
2. log 3 10
3. log
1
1000
4. 10log6
November 13, 2013
Evaluate.
1. ln 4
2. ln e2
3. ln 1
e2
4. ln (-1)
November 13, 2013
Solve.
1. log x = 5
2. ln x = -3
November 13, 2013
Transformations
1. g(x) = log10(x-1)
2. h(x) = 2 + log10x
November 13, 2013
November 13, 2013
3.4 Properties of Logarithmic Functions
Product Rule:
logb(RS) = logbR + logbS
proof:
November 13, 2013
Quotient Rule:
logb R = logbR - logbS
S
November 13, 2013
Power Rule:
logbRc = c logbR
November 13, 2013
Write the following as a sum or difference of logs.
1. log (16x2y3)
x2-3
2. ln
x
November 13, 2013
Write each as a single logarithm.
3. ln x4 - 3 ln (xy)
4.
1
log √x + 3 log (x+1)
10
2 10
November 13, 2013
Change-of-Base Formula:
logbx =
logax
logab
Ex.
1. log35
2. log420
November 13, 2013
Describe how to transform the graph of f(x) = ln x into
the graph of the given function. Sketch the graph.
g(x) = log5x
November 13, 2013
Graph the function, analyze it for domain, range,
continuity, increasing/decreasing behavior, asymptotes,
and end behavior.
f(x) = log1/3(9x)