Question

Which of the following pairs of expression are defined for the same set of values of `x` ? `f_1(x)=2(log)_2xa n df_2(x)=(log)_(10)x^2` `f_1(x)=(log)_xx^2a n df_2(x)=2` `f_1(x)=(log)_(10)(x-2)+(log)_(10)(x-3)a n df_(2(x))=(log)_(10)(x-2)(x-3)dot`

Answer 1

(i)`f_(1)(x) = 2 log_(10)x" is defined for "x gt 0`
` f_(2)(x) = log_(10)x^(2)" is defined for "x^(2) gt 0 or x in R - {0}`
Therefore ` f_(1)(x) and f_(2)(x)` are not defined for same set of values of x.
(ii) `f_(1)(x) = log_(x) x^(2)" is defined for "x gt 0, x ne 1`
`:. f_(1)(x) = 2, x gt 0, x ne 1`
But `f_(2)(x) = 2` is defined for all real x.
Therefore `f_(1)(x) and f_(2)(x)` are not defined for same set of values of x.
(iii) `f_(1)(x) = log_(10)(x-2)+log_(10)(x-3)" is defined if "x-2 gt 0 and x - 3 gt 0`
`:. x gt 3`
`f_(2)(x) = log_(0)(x-2)(x-3)" is defined if "(x-2)(x-3) gt 0 `
` :. x lt 2 or x gt 3`
Therefore ` f_(1)(x) and f_(2)(x)` are not defined for same set of values of x.