Question

Which of the following, when simplified, reduces to unity? (a) `(log)_(10)5dot(log)_(10)20+((log)_(10)2)^2` (b) `(2log 2+ log 3)/(log 48-log 4)` . (c) `-(log)_5(log)_3sqrt(5sqrt(9))` (d) `1/6(log)_((sqrt(3))/2)((64)/(27))`

Answer 1

a)`log_10^5*log_10^2+(log_10^2)^2`
`log_10 10/2 log_10^(10.2)+...`
`(1-log2)(1+log2)+(log2)^2`
`1-(log2)^2+(log2)^2=1`
b)`(2log2+log3)/(log48-log4)=(log4+log2)/(log48-log4)`
`(log4*3)/(log(48/4)=log12/log12=1`
c)`-log_5 log_3 (9^(1/5))^(1/2)`
`-log_5 log_3 3^(2/(5*2))`
`-log_5 (1/5)`
`-log_5 (5)^(-1)`
`-(-1)=1`
d)`1/6log_(sqrt3/2)^(64/27)`
`-1/6log_(sqrt3/2)(27/64)`
`-1/6log_(sqrt3/2)(sqrt3/2)^6`
`-1/6*6=-1`.