Number of solutions of equation
`log_(2)(9-2)^(x)=10^(log10(3-x))`, is

Options

A. 1
B. 2
C. 0
D. None of these

Answer

A

Explanation

We have,
`log_(2)(9-2^(x))=10^(log10(3-x))` is
`implies log_(2)(9-2^(x))=(3-x) " " [:'a^(log_(a)N)=N]`
`implies (9-2^(x))=2^(3)xx2^(-x)`
`implies 9.2^(x)-(2^(x))^(2)=8`
`implies (2^(x))^(2)-9.2^(x)+8=0`
`implies (2^(x)-8)(2^(x)-1)=0`
`implies 2^(x)=2^(3) " or " 2^(x)=2^(0) impliesx=3 "or",x=0`
Also, the given equation exists for
`9-2^(x) gt 0` and `3-x gt 0`
`i.e. "for " 2^(x) lt 9` and `x lt 3`
i.e. `"for" x lt log_(2)9` and `xlt 3`.
Hence, x=0 is the only solution of the given equation.