Suppose that a, b and are positive real numbers such that \(a^{\log_3 7} = 27\), \(b^{\log_7 11} = 49\), and \(c^{\log_{11}25} = \sqrt{11}\). Find the value of \(a^{(\log_3 7)^2} + b^{(\log_7 11)^2} + c^{(\log_{11} 25)^2}\)
+130031
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
