1, if 0 < x < y then loga x < loga y. - Several examples of common logarithms and an exercise with problems to practice applying logarithmic concepts.">1, if 0 < x < y then loga x < loga y. - Several examples of common logarithms and an exercise with problems to practice applying logarithmic concepts.">
Logarithmp65 613 PDF
Logarithmp65 613 PDF
Logarithmp65 613 PDF
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LOGARITHM OF A NUMBER :
The logarithm of the number N to the base 'a' is the exponent indicating
the power to which the base 'a' must be raised to obtain the number N.
This number is designated as loga N.
logaN = x
Hence :
ax
REMEMBER
log102 = 0.3010
log103 = 0.4771
ln 2 = 0.693
ln 10 = 2.303
If a = 10 , then we write log b rather than log10 b .
If a = e , we write ln b rather than loge b .
The existence and uniqueness of the number loga N follows from the properties of an exponential
functions.
From the definition of the logarithm of the number N to the base 'a' , we have an
identity :
log a N
loga M = . loga M
(iv)
logb M =
log a M
log a b
ln a
e = ax
3.
(i)
(ii)
For 0 < a < 1 the inequality 0 < x < y & loga x > loga y are equivalent.
(iii)
0 < x < ap
(iv)
x > ap
(v)
x > ap
(vi)
0 < x < ap
NOTE THAT :
If the number & the base are on one side of the unity, then the logarithm is positive ; If the number &
the base are on different sides of unity, then the logarithm is negative.
The base of the logarithm a must not equal unity otherwise numbers not equal to unity will not have
a logarithm & any number will be the logarithm of unity.
a = a1/n.
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EXERCISEI
Q.1
Let
2
ab (ab) 2 4(a b)
2
2
when a = 43 and b = 57
Q.2(a) If x = log34 and y = log53, find the value of log310 and log3(1.2) in terms of x and y.
(b) If k
log2 5
(log2 5) 2
log b log b N
log b a
Q.4
(b) a
Q.5
Q.6
Q.7
Let a and b be real numbers greater than 1 for which there exists a positive real number c, different
from 1, such that
2(logac + logbc) = 9logabc. Find the largest possible value of logab.
Q.8
Q.9
Q.10
Calculate : 4
5log
81
4 2
3 6 6log8
1
log 9
5
3
409
2
3
.
log 4 (2000) 6 log 5 (2000) 6
3 2
3
log
2
log 25 7
125 log 25 6
Q.11
Simplify :
Q.12
Simplify : 5
Q.13
Q.14
log
log1 / 5 12
4
1
.
log1 / 2
7 3
10 2 21
1log7 2
s2
log 5 4
2 2
= 0.
2
a 2 b5
. Write log2 4 as a function of 's'
s3 1
c
Q.15
Q.16
Given that log2 3 = a , log3 5 = b, log7 2 = c, express the logarithm of the number 63 to the base
140 in terms of a, b & c.
log 2 24 log 2192
Prove that
= 3.
log 96 2 log12 2
Q.17
+5
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Q.18
Q.19
loga b & y =
log3 7
= 27 ; b
log7 11
= 49 and c
log11 25
= 11 . Find the
(log 7 ) 2
(log 11) 2
(log 25) 2
c 11
value of a 3 b 7
.
Q.20
log10 ( x 3)
log10 x 21
2
1
2
(b)
(c) logx2 . log2x2 = log4x2
Q.21
10
xy
If x, y > 0, logyx + logxy =
and xy = 144, then
=
3
2
value of N.
Q.22
Q.23
Q.24
(a)
(b)
(c)
Find the number of positive integers which have the characteristic 3, when the base of the
logarithm is 7.
If log102 = 0.3010 & log103 = 0.4771, find the value of log10(2.25).
(d)
Q.25
(b) 615
&
Let
and
and
logx+1 (x + x 6)2 = 4
Q.4
Q.5
1 log 2 ( x 4)
log
( x 3 x 3)
Q.3
=1
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Q.6
Q.7
1
log 4 + 1 log 3 = log
2x
Q.8
If 'x' and 'y' are real numbers such that, 2 log(2y 3x) = log x + log y, find
Q.9
The real x and y satisfy log8x + log4y2 = 5 and log8y + log4x2 = 7, find xy.
Q.10
Q.11
Q.12
If p = loga bc, q = logb ca, r = logc ab, then prove that pqr = p + q + r + 2.
Q.13
If logba . logca + logab . logcb + logac . logb c = 3 (Where a, b, c are different positive real numbers 1),
then find the value of abc.
Q.14
Let y =
Q.15
Q.16
Q.17
4
4
2 2
Find x satisfying the equation log 2 1 log 2 1
1 .
2 log
x
x4
x 1
Q.18
Solve : log3
Q.19
Prove that :
a
b
2
Q.20
1
1
x 3 27 .
x
.
y
3
log4(x + 2)2 + 3 = log4(4 x)3 + log4(6 + x)3.
2
x 1 = log9 4 x 3 4
(2008) ( x )
x 1
log 2008 x
x2 .
2
= loga b
2
if ba 1
if 1ba
EXERCISEIII
Q.1
Q.2
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ANSWER
SHEET
LOGARITHM
EXERCISEI
Q.1
12
Q.2 (a)
xy 2 xy 2 y 2
,
; (b) 625
2y
2y
1
25
Q.14 2s + 10s2 3(s3 + 1) Q.15
100
2
or 2
Q.16
Q.9 1/6
1 2ac
2c abc 1
1 1 1
Q.21 507 Q.22 (a4, a, a7) or 4 , , 7 Q.23 (a) 0.5386; 1.5386 ; 3.5386 (b) 2058 (c) 0.3522 (d) 343
a a a
(c) 47
Q.25 23040
EXERCISEII
Q.2 x = 1
Q.3 x = 1
Q.20 0 ,
33
7 3 24
,
4
2
EXERCISEIII
Q.1 x = 3 or 3
Q.2
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