Question

If x = log_a (bc), y = log_b (ca), z = log_c (ab) then prove that `1/(1 + x) + 1/(1 + y) + 1/(1 + z)` = 1

Answer 1

Consider `1/(1 + x) = 1/(1 + log_"a""bc")`

= `1/(log_"a""a" + log_"a""bc")`

= `1/(log_"a"("abc")`

= log_(abc) a   ...`[because log_"m""a" = 1/log_"a""m"]`

Similarly, `1/(1 + y)` = log_(abc) b 

`1/(1 + z) = 1/log_(("abc"))"c"`

∴ `1/(1 + x) + 1/(1 + y) + 1/(1 + z)`

= log_(abc) a + log_(abc) b + log_(abc) c

=  log_(abc) [abc]

= 1 ...[∵ log_m m = 1]