Logarithm. Logarithm (Introduction) The logarithmic function is defined as the inverse of the...
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Transcript of Logarithm. Logarithm (Introduction) The logarithmic function is defined as the inverse of the...
Logarithm (Introduction)
xbn log xnb
The logarithmic function is defined as the inverse of the exponential
function. *A LOGARITHM is an exponent. It is the exponent to which the base must
be raised to produce a given number.
For b > 0 and b 1
is equivalent to
Number
Exponent
Base
xnb nxb log
• Examples,
1. Since , then
2. Since , then
3. Since , then
4. Since , then
4162log 4216
4381
8216
3
113
8162log
4813log
13
13log
)(log Ma ,ThenY= .Ma y
Now we solve .Ma y For y, using base-b logarithms:
.Ma y
)(log)(log Ma by
b If:
)(log)(log. May bb
)(log
)(log
a
My
b
b
Changing the base:
IF:
Take the base-b logarithm of each side
Power rule
Divide each side by )(log ab
)(log
)(log)(log
ab
MbMa
Base-change formula:
If a and b are positive numbers not equal to 1 and M is positive, then
Example :
find To four decimal places
• Solution: By using the base-change with a=7 and b=10:
Chick by finding with calculator. Note that we also have
3614.2)7ln(
)99ln()99(log7
)99(log7
3614.27
3614.2)7log(
)99log()99(log7
Common logarithms
• common logarithm is the logarithm with base 10.
• It is indicated by or
sometimes Log(x) with a capital L
• Traditionally, log10 is abbreviated to log.
)(log10 x
Binary logarithm
• In mathematics, the binary logarithm is the logarithm for base 2. It is the inverse function of .
• Domain and range: the domain of the exponential function
is and its range is
Because the logarithm function is the inverse of The domain of is and its range is
n2log
n2
xy 2
),(
),0(
)(log 2 xy xy 2log
),0(
),(
xy 2
Logarithmic equation :
If we have equality of two logarithms with the same base, we use the one-to-one property to eliminate the logarithm.
If we have an equation with only one logarithm,
such we use the definition of logarithm to write and to eliminate the logarithm
NthenMNaMaif ,loglog
yxa )(log
xa y
• Find the solution :
• Solution:xx 32 )1(
xx 3log)1(2log
)3log(.2log)1( xx
)3log(.)2log()2log(. xx
)2log()3log(.)2log(. xx
)2log()3log()2log( x
)3log()2log(
)2log(
x
xx 32 )1(
Original equation
Take log of each side
Power rule
Distributive property
Get all x-terms on one side
Factor out x
Exact solution
4)3(log)3(log 22 xx
4)3)(3(log2 xx
49log 22 x
42 29 x
252 x
5x
Example (2)
Solve
Solution :
4)3(log)3(log 22 xx
Original equation
Product rule
Multiply the binomials
Definition of logarithm
Even root property
To check, first let x=-5 in the original equation :
Because the domain of any logarithmic function is the set of positive numbers, these logarithms are undefined. Now check x=5 in the original equation :
The solution is {5}.
413
42log8log
4)35(log)35(log
22
22
4)8(log)2(log
4)35(log3)5(log
22
22
Incorrect
Correct
• The natural logarithm is a logarithm to base e
• Where e = 2.7182818….
• it is denoted ln x, as ln x = loge x
Natural Logarithm
Reason for being "natural"
The reason we call the ln(x) "natural" :• expressions in which the unknown variable
appears as the exponent of e occur much more often than exponents of 10
• the natural logarithm can be defined quite easily using a simple integral or Taylor series--which is not true of other logarithms
• there are a number of simple series involving the natural logarithm, and it often arises in nature. Nicholas Mercator first described them as log naturalis before calculus was even conceived.
The general definition of a logarithm
Y = ln x means the same as x = ey
And this leads us directly to the following:
• ln 1 = 0 because e0= 1
• ln e = 1 because e1= e
• ln e2= 2 and ln e-3= -3
Properties:
• All the usual properties of logarithms hold for the natural logarithm, for example:
– (where 28 is an arbitrary real number)
• ln (x)a = a ln x
28
logln
loglog 28
x
xxx
e
Proof that d/dx ln(x) = 1/x
F (x) = ln(x)
1. . f ‘ (x) = lim h-->0 (f (x + h) – f (x)) /h• Definition of a derivative
2. = lim h-->0 (ln(x + h) - ln(x))/h• Plugging the function f (x) = ln(x)
3. = lim h-->0 ln( (x + h) /x) /h• Rule of logarithms: log (a) – log (b) = log (a/b)
4. = lim h-->0 ln(1 + h/x)/h• Algebraic simplification: (x + h)/x = 1 + x/h
5. = lim h-->0 ln(1 + h/x) (⋅ x/h) (1/⋅ x)• Algebraically, 1/h = (x/h)(1/x)
6. = 1/x lim ⋅ h-->0 ln(1 + h/x) (⋅ x/h)• 1/x is a constant with respect to the variable being
"limited," so we can pull it out of the limit .
7. = 1/x lim_⋅ h-->0 ln((1 + h/x)x/h)• Rule of logs: log(a) b = log(ab) ⋅• Let's look at a definition of e using a limit:
• e = lim n-->∞ (1 + 1/n)n Or equivalently: e = lim n-->0 (1 + n)1/n
– lim h-->0 (1 + h/x) x/h = e• True from the definition of e (the x is irrelevant,
since it's constant with respect to h)• 1/x lim_⋅ h-->0 ln((1 + h/x)x/h)
8. = 1/x ln(e)⋅• Follows from (7.5) applied to (7)Since e is
the base of ln: ln (e) = 1
9. = 1/x• What happens when you multiply anything
by 1 is that it doesn't change.
Compare between the graphs ofCompare between the graphs of::
xxf 3)(
xxf 31 log)(
xy
xxf 3)( xxf 31 log)(
)b,1(
b1
1
yxb log
xy blog1) for any base ,x-intercept is 1.because the logarithm of 1 is 0
.2) The graph passes through the point (b,1) .because the logarithm of the base is 1.
1log by b
3) The graph is below the x-axis, the logarithm is negative for
10 xWhich number are those that have negative logarithms.
01log by
Ex:(4)Ex:(4)Graph the followingGraph the following
yx
3-/10001
-21/100
-11/10
01
110
2100
31000
)(log1 xo Sol:Sol:
Change to Change to exponential form,exponential form,
yx 10
Ex:(5)Ex:(5)Graph the followingGraph the following
yx
3-/81
-21/4
-11/2
01
12
24
38
)(log 2 x
yx 2
• Sol:Sol:• Change to Change to
exponential form,exponential form,
Ex:(3)Ex:(3)Graph the followingGraph the following..
Sol:Sol:• Change to Change to
exponential exponential form,form,
YX
-30.04
-20.13
-10.36
01
12.71
27.38
320.08
yex
xy ln