If a polynomial function f satisfies the relation #log_2 (f(x)) = log_2 (2+2/3+2/9... oo) . log_3 (1+(f(x))/(f(1/x)))# and #f(10) = 1001# then #f(20) =# is?

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Answer 1 · 2018-06-03T02:26:43

Given relations are

#log_2 (f(x)) = log_2 (2+2/3+2/9... oo) . log_3 (1+(f(x))/(f(1/x)))# and #f(10) = 1001#

when

#log_2 (f(x)) = log_2 [2(1+1/3+1/3^2... oo) ] log_3 (1+(f(x))/(f(1/x)))#

#=>log_2 (f(x)) = log_2 [2xx(1/(1-1/3)) ] log_3 (1+(f(x))/(f(1/x)))#

#=>log_2 (f(x)) = log_2 [2xx(1/(2/3)) ] log_3 (1+(f(x))/(f(1/x)))#

#=>log_2 (f(x)) = log_2 3 log_3 (1+(f(x))/(f(1/x)))#

#=>log_2 (f(x)) = log_2 (1+(f(x))/(f(1/x)))#

#=>f(x) = 1+(f(x))/(f(1/x))#

#=>f(10) = 1+(f(10))/(f(1/10))#

Now inserting #f(10) = 1001#

#=>1001 = 1+1001/(f(1/10))#

#=>f(1/10)=1001/1000=1+1/1000=1+(1/10)^3#

Inserting #1/10=y# we get

#f(y)=1+y^3#

When #y=20#

#f(20)=1+20^3=8001#