To determine which point corresponds to the real zero of the graph of the function [tex]\( y = \log_3(x + 2) - 1 \)[/tex], we must find the value of [tex]\( x \)[/tex] that makes [tex]\( y \)[/tex] equal to zero.
1. Set [tex]\( y \)[/tex] to zero:
[tex]\[
0 = \log_3(x + 2) - 1
\][/tex]
2. Solve the equation for [tex]\( x \)[/tex]:
We start by isolating the logarithmic term:
[tex]\[
\log_3(x + 2) = 1
\][/tex]
Next, we rewrite the logarithmic equation in its exponential form:
[tex]\[
x + 2 = 3^1
\][/tex]
Simplify the right-hand side:
[tex]\[
x + 2 = 3
\][/tex]
Finally, solve for [tex]\( x \)[/tex]:
[tex]\[
x = 3 - 2
\][/tex]
[tex]\[
x = 1
\][/tex]
3. Verify the corresponding [tex]\( y \)[/tex]:
We substitute [tex]\( x = 1 \)[/tex] back into the original function:
[tex]\[
y = \log_3(1 + 2) - 1
\][/tex]
Simplify inside the logarithm:
[tex]\[
y = \log_3(3) - 1
\][/tex]
Since [tex]\( \log_3(3) = 1 \)[/tex]:
[tex]\[
y = 1 - 1
\][/tex]
[tex]\[
y = 0
\][/tex]
We found that the point where the function [tex]\( y = \log_3(x + 2) - 1 \)[/tex] crosses the x-axis (i.e., the zero of the function) is [tex]\( (1, 0) \)[/tex].
Conclusion:
The point [tex]\((1, 0)\)[/tex] corresponds to the real zero of the graph of [tex]\( y = \log_3(x + 2) - 1 \)[/tex].
