Let's determine the [tex]$y$[/tex]-intercept for the logarithmic function [tex]\( f(x) = \log_3(x + 2) + 1 \)[/tex]. To find the [tex]$y$[/tex]-intercept, we need to evaluate the function at [tex]\( x = 0 \)[/tex].
So, we have:
[tex]\[ f(0) = \log_3(0 + 2) + 1 \][/tex]
[tex]\[ f(0) = \log_3(2) + 1 \][/tex]
Now we need to express [tex]\(\log_3(2)\)[/tex] in a more common logarithmic base. We can use the change of base formula for logarithms:
[tex]\[ \log_3(2) = \frac{\log(2)}{\log(3)} \][/tex]
where [tex]\(\log\)[/tex] denotes the natural logarithm.
Using this, the [tex]$y$[/tex]-intercept becomes:
[tex]\[ f(0) = \frac{\log(2)}{\log(3)} + 1 \][/tex]
Thus, the expression that correctly results in the [tex]$y$[/tex]-intercept is:
[tex]\[ \frac{\log 2}{\log 3} + 1 \][/tex]
Therefore, the correct answer is:
[tex]\[ \frac{\log 2}{\log 3} + 1 \][/tex]
