The number of solutions of the equation
`log_(3)(3+sqrt(x))+log_(3)(1+x^(2))=0`, is

Options

A. 0
B. 1
C. 2
D. more than two

Answer

A

Explanation

Clearly, the given equation is meaningful for `x ge0`. We observe that
`3+sqrt(x) gt 3` and `1+x^(2) gt 1` for all `x gt 0`
`implies log_(3)(3+sqrt(x)) gt 1` and `log_(3)(1+x^(2)) gt 0` for all ` x gt 0`
`implies log_(3)(3+sqrt(x))+log_(3)(1+x^(2)) gt 0 ` for all ` xge0`
Hence, the given equation has no solution.