Option A: [tex]x+1=3^{2}[/tex] is the correct statement
Explanation:
The expression is [tex]\log _{3}(x+1)=2[/tex]
We need to determine the statement which is true for the expression [tex]\log _{3}(x+1)=2[/tex]
Using logarithmic definition, if [tex]\log _{a}(b)=c[/tex] then [tex]b=a^{c}[/tex], we have,
[tex]x+1=3^{2}[/tex]
Simplifying,
[tex]x+1=9[/tex]
[tex]x=8[/tex]
Now, we shall determine the statement which is true for the expression [tex]\log _{3}(x+1)=2[/tex]
Option A: [tex]x+1=3^{2}[/tex]
Simplifying, we get,
[tex]x+1=9[/tex]
[tex]x=8[/tex]
Thus, the expression [tex]x+1=3^{2}[/tex] is equivalent to [tex]\log _{3}(x+1)=2[/tex]
Hence, Option A is the correct answer.
Option B: [tex]x+1=2^3[/tex]
Simplifying, we get,
[tex]x+1=8[/tex]
[tex]x=7[/tex]
Since, the values of x from both the expressions are not equal.
Then, the expression [tex]x+1=2^3[/tex] is not equivalent to [tex]\log _{3}(x+1)=2[/tex]
Hence, Option B is not the correct answer.
Option C: [tex]2(x+1)=3[/tex]
Simplifying, we get,
[tex]2x+2=3[/tex]
[tex]2x=1[/tex]
[tex]x=\frac{1}{2}[/tex]
Since, the values of x from both the expressions are not equal.
Then, the expression [tex]2(x+1)=3[/tex] is not equivalent to [tex]\log _{3}(x+1)=2[/tex]
Hence, Option C is not the correct answer.
Option D: [tex]3(x+1)=2[/tex]
Simplifying, we get,
[tex]3x+3=2[/tex]
[tex]3x=-1[/tex]
[tex]x=-\frac{1}{3}[/tex]
Since, the values of x from both the expressions are not equal.
Then, the expression [tex]3(x+1)=2[/tex] is not equivalent to [tex]\log _{3}(x+1)=2[/tex]
Hence, Option D is not the correct answer.
