LOGARITHMS
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1) CONVERT in to logarithmic form:
a) 6⁻¹ = ⅙ b) ³√(27) =3
c) √72=6√2
2) convert in exponential form
a)log₅(625) = 4 b) log₄(4) = 1
3)Simplify->
a) log₃(81) b) log₁₀³√(100)
c) log₂(1/32) d) log₉(27)
e) log₇343 f) log₅√₅125
g) log₃log₄log₃81 h)log₃5 .log₂₅27
i) 1/2log(25)- 2log(3)+log(18)
j) log2+16log(16/15)+
12log(25/24) + 7log(81/80)
k) log(81/8) - 2log(3/2) + 3log(⅔)
+ log(3/4)
l) 7log(16/15)+5log(25/24)
+3log(81/80)
m) {log√(27)+log(8)+log√(1000)}/
log(120)
n) log₂(10) - log₁₆(625)
o) log₃log₂log₂(2⁸)
p) 23log(16/15)+17log(25/24)
+ 10log(81/80)
q) logᵤa . logᵥx . logₐv
r) log₈√[8 {√8√(8).....∞}]
s) log₃[⁴√{(729) . ³√(9⁻¹) .27⁻⁴/³}]
t) {log√(27)+log√(8)+log√(125)}/
{ log(6) + log(5)}
4)Find x
a) log₃(x)= 4 b) log₂₅(x)=-1/2
c) log₁/₂(x)= -3 d) x=log₁₀(0.001)
e) log₂₅(x) = -½ f) logₓ(243) = 5
5)Compute the base of following
a) log 9= 2 b) log 2 = - 3
c) log 324 =4
6) Find the logarithm of 144 to the base 2√(3)
7) The logarithm of a number to the base √(2) is a, what is its logarithm to the base 2√(2)
8) If log(x²y³) =a and log(x/y)= b find log x and lig y in terms of a and b.
9) Express in logarithm form of individual letters
logₑ ³√{(a.b³)/(c¹/².d)}
10) If log 2= 0.30103 find
log 2000.
11)) If log₁₀2=0.3010,
log₁₀3 =0.7781, log₁₀7= 0.8451 prove
a) log₅(5)=0.6990
b) log₁₀(45)= 1.6532
c) log₁₀(2.4)=
d) log₁₀(6)=0.7781
e) log₁₀(108)=2.0333
f) log₁₀³√(5) = 0.2330
g) log₁₀(70) = 1.8451
h) log84 =1.9242793
i) log21.6= 1.3344539
j) log(0.00693)= ⃗3.840733.
k) log294= 2.4683473
l) log√(4.5)= 0.3266063
12) If log₁₀2=0.3010 , log₁₀3=0.4771,
show log₁₂40= 1.485
13) If log₁₀3= 0.4771 find
a) log₂₅125 b) log₁₀3000
13)i) if log(2)=.3010 and
log(3)=.4771 find
a)log(8) b) log(24)
c) log(108) d) log(25)
e) (.405)¹/²
f) [(2.7)³ . (.81)⁴/³ / (90)⁴/⁵}
14)Prove
a) log(75/16) - 2log(5/9)
+log(32/243) =2.
b) 7log(10/9)-2log(25/24)+
3log(81/80) = log(2)
c) 16log(16/15)+12log(25/24)
+ 7log(81/80) = log(5)
d) {log√7+log 8- log√1000}/
log(1.2) = 3/2
e) log2+16log(16/15) +log(25/24)
+7log(81/80) = 1
f) log(11/15)+ log (490 /297)
- 2log(7/9) =log(2)
g) log(81/8) - 2log(3/2)+3log(2/3)
+ log(3/4) = 0
h) log(36/25)² + 3log(2/9) -log(2)
=2log(16/125)
i) log₃log₂log₂(256) = 1
j) log₂log√₂log₃(81) = 2
k) log₂[log₂{log₃(log₃27³}] =0
l) 1/6√{(3log 1728)/
(1+1/2 log0.36+1/8 log8)}=1/2
m) logₓa . logᵥx= logᵥa
n) (1/logₐavx)+ (1/logᵥavx) +
(1/logₓavx) =1
o) logₐx² . logᵥa³ . logₓv⁴ = 24
p) (log√27 + log√8- log√125)/
(log 6 - log 5) = 3/2
q) log₃[√{3√(3)......∞}]=1
15) If x²+y² = 6xy, prove
2log(x+y) = logx + log y +3 log 2
16) If a² +b² =23ab prove
log(a+b)/5 = 1/2(log a+ log b).
17) If a³⁻ˣ.b⁵ˣ=aˣ⁺⁵.b³ˣ, prove
x log(b/a) = log(a)
18) If a²+b²=14ab, prove
log{(a+b)/2}= 1/2 (log a + log b)
19) If a²+b² = 27ab prove
log{(a - b)/5 }= 1/2 (log a + log b)
20) If log{(a+b)/3}
=1/2(log a+ log b)
prove a/b + b/a =7
21) If log{(a-b)/4} =
1/2 (lig a+ log b)
prove a² + b² = 18ab.
22) If a²+b²= 7ab prove log (a+b) =
½(loga +logb).
23) If x² + y² =11xy prove
2log(x-y)=2log3 + log(x) +log(y)
24) If a² =b³=c⁵ =x⁶ then prove
logₓ(abc) = 31/5.
25)If log(a+b)/7=
1/2 {log(a) +log(b)} then prove
a/b +b/a = 47
26)solve
a)log₂log₃log₂(x) = 1
b) logₓ(8x - 3) - logₓ(4) = 2
c) ½ log(11+4√(7)) = log(2+x)
d) {log₁₀(x-5)}/2 +{13-log₁₀(x)}/3=2
e) log₃(3+x)+ log₃(8-x)- log₃(9x-8) =
2 - log₃(9)
f) 5ˡᵒᵍ ˣ + 3ˡᵒᵍ ˣ = 3logˡᵒᵍ ˣ⁺ ¹ -
5logx -1
g) log₅(5¹/ˣ) + 125 = log₅(6) + 1
+1/2x
h) 1/(logₓ 10) +2 = 2/(log₀,₅10)
i) logₓ(2).logₓ/₁₆(2) = logₓ/₆₄(2).
j) log₂x+log₄x+log₁₆x=21/4
27)If log x/(y-z)=log y/(z-x)=logz/(x-y) then prove xyz=1
28) (yz)ˡᵒᵍʸ⁻ˡᵒᵍᶻ (zx)ˡᵒᵍᶻ⁻ˡᵒᵍˣ
(xy)ˡᵒᵍˣ⁻ˡᵒᵍʸ =1
29)If log a/(w-r)=log b/(r-p)=
logc/(p-q) prove aᵖbʷcʳ.
30) If a=b²=c³=d⁴, prove
logₐ(abcd)=25/12
31) If a,b,c are any three consecutive positive integers, prive that log(1+ac) = 2log(b)
32) prove without using log table that, log₁₀2 > 0.3.
33) If x= 1+logₐvx, y=1+logᵥxa,
z= 1+logₓav then show
xy+yz+zx=xyz
34) If x= logₐvx, y=logᵥxa,z=logₓav then show x+y+z+2=xyz
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