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Properties of Logarithms
1. loga 1 = 0 since a0 = 1.
2. loga a = 1 since a1 = a.
4. If loga x = loga y, then x = y. one-to-one property
3. loga ax = x and alogax = x inverse property
log6 6 = 1 property 2→ x = 1
log3 35 = 5 property 3
Examples:Solve for x: log6 6 = x
Simplify: log3 35
Simplify: 7log79 7log79 = 9 property 3
3
The graphs of logarithmic functions are similar for different values of a. f(x) = loga x (a 1)
3. x-intercept (1, 0)
5. increasing6. continuous7. one-to-one 8. reflection of y = a x in y = x
1. domain2. range
4. vertical asymptote
Graph of f (x) = loga x (a 1)
x
y y = x
y = loga x
y = a x
domain
range
y-axisvertical
asymptote
x-intercept(1, 0)
4
The graphs of logarithmic functions are similar for different values of a. f(x) = log4 x
x
y
domain
range
y-axisvertical
asymptote
x = 4 y
y = log4 x
x-intercept(1, 0)
y = x
log4 x = y 4 y = x
5
Shifting Graph of Logarithmic Function f(x) = log4 (x-1)
x
y
domain
range
y = log4 x
y = x
x-intercept(2,0)
y = log4 x-1
6
Shifting Graph of Logarithmic Function f(x) = 2+log4x
x
y
domain
range
y = log4 x
y = x+3
(1,2)
y = 2 + log4 x
1
2
Properties of Logarithm
ExamplesExpand: 1. log3(2x) = log3(2) + log3(x)
2. log4( 16/x ) = log4(16) – log4(x)
3. log5(x3) = 3log5(x)
4. log2(8x4) – log2(5) = log2(8) + log2(x4) – log2(5)
ExamplesCondense: 1. log2(x) + log2(y) = log2(xy)
2. log3(4) – log3(5) = log3(4/5)
3. 3log2(x) – 4log2(x + 3) + log2(y) log2(x3y) log2((x + 3)4)
Natural Logarithm and e
is used to denote
€
y =lnx
€
y =loge x
€
y =ln(x)
€
f(x) =e x
€
f−1(x) =ln(x)then
€
y =e x
€
y =x
Using the natural log - ln
Evaluate the following ln’s:
€
ln1
lne
lne 2
ln e
ln1e
€
e 0 =1
Without using a calculator find the value of:
€
lne 3
lne 4
ln e3
ln 1e 3
lne n
= 3
= 4
=
€
13
= -3
= n
=0
=1
=2
= -1
=
€
12
The laws of natural logarithms
€
lna + lnb =lnab
lna−lnb =lnab
lnab =b lna
Find x, if
€
e x =20
Remember the base of a natural log is e.
€
lne x =8
Rearrange in index form.
€
loga b =c ⇔ b =ac
€
x =e 8
€
x =2980 .96
€
lnx =10
lnx =4
lnx =0 .5
Find x in each of the following:
€
x =22026
€
x =54 .6
€
x =1 .65
Take a natural log of both sides.
Find x, if
€
lnx =8
€
lne x =ln20
Use the power rule.
€
x lne =ln20
€
x =ln20
€
x =3
€
e x =100
e x =3500
e x =0 .25
€
x =4 .61
€
x =8 .16
€
x =−1 .39
The graph of exponential funtion
The graph of f(x) = ax, a > 1
Domain: (–∞, ∞)
Range: (0, ∞)
Horizontal Asymptote y = 0
y
x4
4
(0, 1)
The graph of
f(x) = ax, a > 1
is INCREASING
over its domain
Example 3
x
y
2–2
4
f(x) = 2x
Example: Sketch the graph of f(x) = 2-x. State the domain and range.
Domain: (–∞, ∞)
Range: (0, ∞)
Transformation of exponential graphs
Example: Sketch the graph of f(x) = 2 x + 2
State the domain and range.
x
y
y =0
2
4
Domain: (–∞, ∞)
Range: (0, ∞)
Range: (0, ∞)
f(x) = 2 x + 2
f(x) = 2x ( 22 )
f(x) = 2x (4)
(0, 4)
f(x) = 2x
f(x) = 2x=2
Example 3
x
y
2–2
4
f(x) = 2x
Example: Sketch the graph of f(x) = -2x. State the domain and range.
Domain: (–∞, ∞)
Range: (0, -∞) f(x) = -2x
Graph of f(x) = ex y
x2 –2
2
4
6 x f(x)-2 0.14-1 0.380 11 2.722 7.39
The irrational number e, where
e ≈ 2.718281828…
is used in applications involving growth and decay.
20
The graphs of logarithmic functions are similar for different values of a. f(x) = log4 x
x
y
domain
range
y-axisvertical
asymptote
x = 4 y
y = log2 x
x-intercept(1, 0)
y = x
log4 x = y 4 y = x
21
Shifting Graph of Logarithmic Function f(x) = log4 (x-1)
x
y
domain
range
y = log4 x
y = x
x-intercept(2,0)
y = log4 x-1
22
Shifting Graph of Logarithmic Function f(x) = 2+log4x
x
y
domain
range
y = log4 x
y = x+3
(1,2)
y = 2 + log4 x
1
2
€
y =ln(x)
€
f(x) =e x
€
f−1(x) =ln(x)then
€
y =e x
€
y =x
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