Download a function? - Old Saybrook Public Schools

January 27, 2017
3.2 Logarithmic functions and their graphs
Does y = 2x have an inverse that is
a function?
Yes. The graph passes
the vertical line test
Inverse: x = 2y
or
Both equations express
the same relationship
between x, y, and 2.
The base on the exponent is the
same as the base on the
logarithm.
A logarithm gets the exponent
alone
January 27, 2017
Write in log form.
Write in exponential form.
1.)
73=243 log7243=3 7.) log381 = 4
2.)
82=64
log864=2
3.)
53=125 log5125=3
4.)
ax = 9
5.)
ab = c
6.) ex=y
8.)
9.)
log2 32 = 5
25=32
3
log4 x = 3 4 =x
loga9=x
logac=b
logey=x
10.) logb 10 = a
ba=10
11.) logx a = z
xz=a
January 27, 2017
Write in log form.
1.)
73=243
log7 243 = 3
2.)
82=64
log8 64=2
3.)
53=125
log5 125 = 3
4.)
ax = 9
loga 9 = x
5.)
ab = c
loga c = b
6.) ex=y
loge y = x
written as ln y = x loge is
called the natural logarithm
Write in exponential form.
7.) log381 = 4
34 = 81
8.)
log2 32 = 5
25 = 32
9.)
log4 x = 3
43 = x
10.) logb 10 = a
ba = 10
11.) logx a = z
xz = a
January 27, 2017
January 27, 2017
January 27, 2017
The natural log function
y = ln x
means y = loge x
inverse of y = ex
The common log function y = log10 x
also written as y = log x
Evaluate on a calculator: log1020
log 5
ln 9
ln 40
Evaluate without a calculator
1.)
2.)
3.)
4.)
5.)
6.)
January 27, 2017
Graphs of log functions
inverses
Increases rapidly
increases slowly w/o bound
without bound
(no horizontal asymptote)
Decreasing
exponential function
Decreasing logarithmic
function
0<b<l
Or
0<b<1
Or
y = 2-x
y = -log2 x
January 27, 2017
Draw an accurate graph for each. State the
domain, equation of the asymptote and the
y = log3x
x-intercept.
y = log2(x-1)
y = log2 x + 3
Graph each function.
y = log3x
y = log2(x-1)
D: x > 1
v.a. x = 1
x-int: (2, 0)
y = log2 x + 3
(x, y+3)
x
1/4
1/2
1
2
4
y
D: x > 0
1
v.a. x = 0
2
3
4
5
x-int: (1/8, 0)
To find x-int
0 = log2x + 3
-3 = log2x
2-3 = x
1/8 = x
January 27, 2017
Graph each function.
y = log3x
y = log2(x-1)
y = log x + 2
Find the domain, vertical asymptote and
x-intercept for each function.
1.) y = ln (x+2)
3.) y = ln x2
2.) y = ln (2-x)
January 27, 2017
Find the domain, vertical asymptote and xintercept for each function.
1.) y = ln (x+2)
2.) y = ln (2-x)
D: x > -2
v.a. x = -2
x-int: (-1,0)
D: x < 2
v.a. x = 2
x-int: (1,0)
reflect
y = ln(-x)
3.) y = ln x2
ar
m
gu
ca
en
n
t
t
no
0
beD: x ≠ 0
v.a. x = 0
x-int: (1,0) (-1,0)
reflect over y-axis,
then shift right 2
now right 2
y = ln-(-2+x)
January 27, 2017
Find the domain, vertical asymptote and xintercept for each function.
1.) y = -log3(x-3)
2.) y = ln(-x-5)
3.) y = 2log x - 4
Find an equation for the inverse of each function.
1.) y = 5x
2.) y = log4 x
3.) y = ln x
4.) y = 10x
January 27, 2017
Find an equation for the inverse of each function.
1.) y = 5x
2.) y = log4 x
y = 4x
y = log5x
3.) y = ln x
4.) y = 10x
y = ex
y = log x
Solve each equation
10x = 125.893
Switch to log form:
log2 x = 7
Switch to exponential form
27=X
Evaluate on calculator:
2.1 ≈ x
January 27, 2017
Solve each equation. Show work.
1.) log7 x = log7 5
2.) log5 x = 3.1
3.) log5 54 = x
4.) 10x= 80
5.) ex = 17
6.) log8 3x = log8 (2x+5)
Solve each equation. Show work.
1.) log7 x = log7 5
x=5
3.) log5 54 = x
x=4
5.) ex = 17
x ≈ 2.833
2.) log5 x = 3.1
53.1 =
x ≈ 146.827
4.) 10x= 80
x ≈ 1.903
6.) log8 3x = log8 (2x+5)
x=5
January 27, 2017