Question

Show that 1-:logx to the base 2 +1-:logx to the base 3 +........+1-:logx to the base 43 =1-:log x to the base 431 ?

Answer 1

The result is not `1-:log_431x` but `1:-log_(43!)x`

Explanation:

To simplify `1-:log_2x+1-:log_3x+1-:log_4x........+1-:log_43x`

we can use the identity that `log_abxxlog_ba=1` or `1/log_ab=log_ba`

Hence `1-:log_2x=log_x2`, `1-:log_3x=log_x3` and so on.

Hence `1-:log_2x+1-:log_3x+1-:log_4x........+1-:log_43x`

= `log_x2+log_x3+log_x4+....+log_x43`

= `log_x(2xx3xx4xx.....xx43)`

= `log_x43!`

= `1-:log_43!x`

It is obvious that answer is not `1-:log_431x` but `1-:log_(43!)x`