Suppose that a, b and are positive real numbers such that alog37=27, blog711=49, and clog1125=√11. Find the value of a(log37)2+b(log711)2+c(log1125)2
+130475
By the change-of-base theorem.....
log3 7 = log 7 / log 3
So
a^(log 7 /log 3) = 27
So
a^(og3 7)^2 =
a^ [ (log 7/log3) * (log7 /log3 ) ] = 27^(log7/log3)
Similarly
b^[ log 11/ log 7]^2 =
b^[ (log 11/log 7) *(log11/log 7) ] = 49^(log11/log 7)
And
c^[(log25/log11) * (log25/log11)] = sqrt(11)^(log25/log11)
So
27^(log7/log3) + 49^(log11/log7) + sqrt(11)^(log 25/log11) =
27^(log3 7) + 49^(log7 11) + sqrt (11)^(log11 25) =
(3^3)^(log 3 7) + (7^2)^(log 7 11 ) + (11^[1/2] )^(log11 25) =
(3 ^log3 7)^3 + (7 ^ log 7 11)^2 + ( 11^log11 25 )^(1/2)
By a log property.....a^(log a b ) = b
So we have
(7)^3 + 11^2 + 25^(1/2) =
343 + 121 + 5 =
469
