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1.If x,a and m are any three numbers connected by the relation:
m=a^{x } (a>0, a≠1), then,
"x" is defined as the logarithm of "m" to the base "a" and is written as:
x= log_{a} m
2.Some important results:
(a) m =a ^{log am}
(b) x =log _{a} (a^{x})
(c) log _{a} 1 = 0
3.Some important theorems:
(a) log _{a} (mn) = log _{a }m + log _{a} n
(b) log_{ a} (m/n) = log_{ a }m  log _{a }n
(c) log_{ a }(m^{n}) = n. log _{a }m
(d) log _{a} m = (log _{b} m) / (log _{b} a) ……. Change of base theorem
(e) log_{ a }a = 1
(f) log_{ a }b * log_{ b} a = 1
1.If a^{x} = b^{y}, then
a.log a/b= x/y b.log a/ log b = x/y c.log a/ log b = y/x d.log b/a = x/y
2.2 log_{10} 5 + log_{10} 8  ½ log_{10} 4 = ?
a.2 b.4 c.2 + 2 log_{10} 2 d.4  4 log_{10} 2
3.log_{a} (ab) = x, then log _{b} (ab) is :
a.1/x b.x/ (x+1) c.x/(1x) d.x/(x1)
4.If log_{8} x + log _{8} 1/6 = 1/3, then the value of x is:
a.12 b.16 c.18 d.24
5.The value of (log_{9} 27 + log_{8} 32) is:
a.7/2 b.19/6 c.5/3 d.7
6.If log_{12} 27 = a, then log_{6 }16 is:
a.(3a)/4(3+a) b.(3+a)/4(3a) c.4(3+a)/(3a) d.4(3a)/(3+a)
7.The value of (1/log_{3} 60 + 1/log_{4} 60 + 1/log _{5} 60) is:
a.0 b.1 c.5 d.60
8.If log x + log y = log (x+y), then,
a.x=y b.xy=1 c.y= (x1)/x d.y=x/(x1)
9.If log 27= 1.431, then the value of log 9 is:
a.0.934 b.0.945 c.0.954 d.0.958
10.If log 2= 0.030103, the number of digits in 2^{64} is :
a.18 b.19 c.20 d.21
Answer & Explanations
1.(c). a^{x} = b^{y} => log a^{x} = log b ^{y} => x log a = y log b
=> log a/ log b = y/x
2.(a). 2 log_{10} 5 + log_{10} 8  ½ log_{10} 4
= log_{10} (5^{2}) + log_{10} 8  log_{10} (4^{1/2})
= log_{10} 25 + log_{10 }8  log_{10} 2 = log _{10} (25*8)/2
= log_{10} 100 = 2
3.(d). log_{a} (ab) = x => log b/ log a = x => (log a + log b)/ log a = x
1+ (log b/ log a) = x => log b/ log a = x1
log a/ log b = 1/ (x1) => 1+ (log a/ log b) = 1 + 1/ (x1)
(log b/ log b) + (log a/ log b) = x/ (x1) => (log b + log a)/ log b = x/ (x1)
=>log (ab)/ log b = x/(x1) => log_{b} (ab) = x/(x1)
4.(a). log_{8} x + log_{8} (1/6) = 1/3
=> (log x/ log 8) + (log ^{1}/_{6} / log 8) = log (8 ^{1/3}) = log 2
=> log x = log 2  log 1/6 = log (2*6/1)= log 12
5.(c). Let log_{9} 27 = x. Then, 9^{x} = 27
=> (3^{2})^{x} = 3^{3} => 2x = 3 => x= 3/2
Let log_{8} 32 = y. Then
8^{y} = 32 => (2^{3})^{y} = 2^{5} => 3y = 5 => y=5/3
6.(d). log_{12} 27 = a => log 27/ log 12 = a
=> log 3^{3} / log (3 * 2^{2}) =a
=> 3 log 3 / log 3 + 2 log 2 = a => (log 3 + 2 log 2)/ 3 log 3 = 1/a
=> (log 3/ 3 log 3) + (2 log 2/ 3 log 3) = 1/3
=> (2 log 2)/ (3 log 3) = 1/a  1/3 = (3a)/ 3a
=> log 2/ log 3= (3a)/3a => log 3 = (2a/3a)log2
log_{16} 16 = log 16/ log 6 = log 2^{4}/ log (2*3) = 4 log 2/ (log 2 + log 3)
= 4(3a)/ (3+a)
7.(b). log_{60} 3 + log_{60} 4 + log_{60 }5 + log _{60} (3*4*5)
= log_{60} 60 = 1
8.(d). log x + log y = log (x+y)
=> log (x+y) = log (xy)
=> x+y = xy => y(x1) = x
=> y= x/(x1)
9.(c). log 27 = 1.431 => log 3^{3} = 1.431
=> 3 log 3= 1.431 => log 3 = 0.477
Therefore, log 9 = log 3^{2} = 2 log 3 = (2*0.477) = 0.954
10.(c). log 2^{64} = 64 log 2 = (64*0.30103) = 1926592
Its characteristics is 19.
Hence, the number of digits in 2^{64} is 20


























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