If `log_2 10=P ;(log_e 10)/(log_e7)=q` and `(11)^r= 10`, then which one of the following expressions is equivalent to `log_(10) 154` ?

Options

A. pqr
B. `(1)/(pqr`
C. `(p+q+r)/(pqr)`
D. `(pq+qr+rp)/(pqr)`

Answer

D

Explanation

`log_(10)154=log_(10)2+log_(10)7+log_(10)11`
Now `log_(2)10 = p`
`rArr (1)/(log_(10)2)=p`
`rArr log_(10)2=(1)/(p)` ….(1)
`(log_(e )10)/(log_(e )7)=q`
`rArr (1)/(log_(10)7)=q`
`rArr log_(10) 7=(1)/(q)` ….(2)
And `11^(r ) = 10`
`rArr r log_(10)11=1`
`rArr log_(10)=(1)/(r )` ....(3)
`therefore log_(10)154 =(1)/(p)+(1)/(q)+(1)/(r )=(pq+qr+rp)/(pqr)`