To determine which point corresponds to the real zero of the graph of [tex]\( y = \log_3(x + 2) - 1 \)[/tex], we need to find the value of [tex]\( x \)[/tex] that makes [tex]\( y = 0 \)[/tex].
Given the equation:
[tex]\[ y = \log_3(x + 2) - 1 \][/tex]
Set [tex]\( y \)[/tex] to 0, since we are looking for the real zero:
[tex]\[ 0 = \log_3(x + 2) - 1 \][/tex]
Now, solve for [tex]\( x \)[/tex]:
First, isolate the logarithmic term:
[tex]\[ \log_3(x + 2) = 1 \][/tex]
To remove the logarithm, recall the definition of a logarithm: if [tex]\(\log_b(a) = c\)[/tex], then [tex]\(b^c = a\)[/tex]. Here, our base is 3, and our logarithm equals 1:
[tex]\[ 3^1 = x + 2 \][/tex]
Evaluate the exponent:
[tex]\[ 3 = x + 2 \][/tex]
Now, solve for [tex]\( x \)[/tex]:
[tex]\[ x = 3 - 2 \][/tex]
[tex]\[ x = 1 \][/tex]
So, the real zero of the function occurs at [tex]\( x = 1 \)[/tex]. This gives us the point:
[tex]\[ (1, 0) \][/tex]
Therefore, the point which corresponds to the real zero of the graph of [tex]\( y = \log_3(x + 2) - 1 \)[/tex] is [tex]\((1, 0)\)[/tex].
The correct answer is: [tex]\((1, 0)\)[/tex].
