Logarithmic Functions and
Their Graphs (II)
Properties of Logarithms
Graphs of Logarithm Functions
Finding Domains of Log Functions
From the definition of a logarithm we know . ….
b = x equals log b x = y
y
Substituting the second equation’s value for y
into the first equation, we get …….
b
logb x
=x x>0
So for the typical bases used you get …………
10
log x
= x and e
ln x
=x
x>0
So you have the following:
Basic Properties of Logarithms:
Suppose that M , N, and b are positive
real numbers, where b ≠ 1, and r is any
real number. Then we have.......
Product Rule :
log b MN = log b M + log b N
Basic Properties of Logarithms:
Product Rule :
log b MN = log b M + log b N
examples of product rule :
log 2 16 = log 2 8 + log 2 2
log 2 16 = log 2 4 + log 2 4
Basic Properties of Logarithms:
Suppose that M , N, and b are positive
real numbers, where b ≠ 1, and r is any
real number. Then we have.......
Quotient Rule :
M
log b
= log b M − log b N
N
Basic Properties of Logarithms:
Quotient Rule :
M
log b
= log b M − log b N
N
examples of quotient rule :
log 2 16 = log 2 32 − log 2 2
log 2 8 = log 2 32 − log 2 4
Basic Properties of Logarithms:
Suppose that M , N, and b are positive
real numbers, where b ≠ 1, and r is any
real number. Then we have.......
Power Rule :
log b N = r log b N
r
Basic Properties of Logarithms:
Power Rule :
log b N = r log b N
r
examples of power rule :
log 2 16 = 4
log 2 16 = log 2 4 2 = 2 log 2 4 = 4
The Power Rule is crucial because it
lets us directly solve exponential
functions.
17 x = 367
one-to-one function, so take the log of both sides
log17 x = log 367
x log17 = log 367
log 367
x=
≈ 2.084333628
log17
The Power Rule is crucial because it
lets us directly solve exponential
functions.
Power Rule :
log b N r = r log b N
Also from the Power Rule we get the following results …..
log b b = r
r
log10 r = r and ln er = r
Problems using the basic properties of Logs:
Writing expressions as sums & differences of logs
(
)
log x 7 y11 =
⎡ (y + 7)3 ⎤
ln ⎢
⎥=
y ⎦
⎣
(
)
log x 7 y11 = log x 7 + log y11
(
)
log x 7 y11 = 7 log x + 11(log y)
1
⎡ (y + 7)3 ⎤
1
3
2
ln ⎢
⎥ = ln(y + 7) − ln y = 3ln(y + 7) − ln y
2
y ⎦
⎣
Problems using the basic properties of Logs:
Combining sums and differences of logs
⎛ x2 ⎞
2 ln x − 3ln y = ln ⎜ 3 ⎟
⎝y ⎠
2
⎡
(c
− cd) ⎤
log(c 2 − cd) − log(2c − 2d) = log ⎢
⎥
(2c
−
2d)
⎣
⎦
⎡ (c 2 − cd) ⎤
⎡ c(c − d) ⎤
⎛ c⎞
= log ⎢
= log ⎢
= log ⎜ ⎟
⎥
⎥
⎝ 2⎠
⎣ 2(c − d) ⎦
⎣ (2c − 2d) ⎦
Change of Base Formula: Very helpful to change to common or natural logs from
other bases
If you forget the formula it is easily derived.
1. Start with the definition of the log, then change to
exponential form.
2. Now, take the log of each side of the equation.
3. Solve for y.
The text’s book’s basic properties of logarithms…..
1. Put in logarithmic form
2. Exchange the x and y variables
Symmetric about y = x line, property of inverse functions
Log graphs contain the point (1,0)
General shape of LOG graph where the base is greater
than 1
The log with
the smaller
base will be
on top.
Think Why.
More accurately,
treats the y-axis as
a vertical
asymptote
General shape of LOG graph where the base is between 0
than 1
The log with
the larger
base will be
on top above
the x-axis and
below
beneath the xaxis.
Think Why.
Transformations of LOG functions:
Transformations of LOG functions:
In General Terms …….
Exponential _ Function :
f (x) = 4 x
⎛ 1⎞
f (x) = ⎜ ⎟
⎝ 3⎠
f (x) = 10 x
f (x) = e x
x
Exponential _ Function :
f (x) = 4 x
⎛ 1⎞
f (x) = ⎜ ⎟
⎝ 3⎠
x
f (x) = 10 x
f (x) = e x
Domain exponential function = Range logarithmic
function
Range exponential function = Domain logarithmic
function
so, Logarithmic _ function _ Domain → x _ in _ int erval _(0, ∞)
Range → y _ in _ int erval _ ( −∞, ∞ )
That does not mean the domain is
always 0 to infinity, though.
1
⎛1 ⎞
2x − 1 > 0, so _ x > , Domain _ is _ thus _ ⎜ , ∞ ⎟
⎝2 ⎠
2
Problems for you …..
From pages 203 and 204
#39-65 odds