We use cookies to personalise content and advertisements and to analyse access to our website. Furthermore, our partners for online advertising receive pseudonymised information about your use of our website. cookie policy and privacy policy.
 
+0  
 
0
110
1
avatar

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 \(a^{(\log_3 7)^2} + b^{(\log_7 11)^2} + c^{(\log_{11} 25)^2}\)
 

 Jan 2, 2019
 #1
avatar+101870 
+1

 

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

 

 

cool cool cool

 Jan 2, 2019
edited by CPhill  Jan 2, 2019

6 Online Users

avatar