Verify the logarithm

Question

(log_a(bc))/(1 + log_a(bc)) + (log_b(ca))/(1 + log_b(ca)) + (log_c(ab))/(1 + log_c(ab)) = 2

Answer 1

log_a(bc)/(1 + log_a(bc)) + log_b(ca)/(1 + log_a(ca)) + log_c(ab)/(1 + log_c(ab))

First Term

 log_abc/(1 + log_a(bc)) = log_abc/(log_aa + log_a(bc)) = log_abc/( log_a(abc)) = log(bc)/log (abc)

Second Term

log_bca/(1 + log_b(ca)) = log_bca /(log_bb + log_a(ca)) = log_bca /( log_b(abc)) = log(ca)/log (abc)

Third Term

log_cab/(1 + log_c(ab)) = log_cab /(log_cc + log_c(ab)) = log_cab /( log_c(abc)) = log(ab)/log (abc)

Adding all three terms

log(bc)/log (abc) + log(ca)/log (abc) + log(ab)/log (abc) = log (bccaab)/ log (abc)

=log(abc)²/log(abc) = 2 log(abc)/log(abc)

=2