A student was asked to determine the $y$-intercept for the logarithmic function $f(x)=\log _{3}(x+4)+2$. Which of the following expressions would result in the correct $y$ intercept?
$3^{-2}-4$
$(-2)^{3}-4$
$\frac{\log 3}{\log 4}+2$
$\frac{\log 4}{\log 3}+2$
Answer
\(\boxed{3.26}\) is the final answer.
Step 1 ：The y-intercept of a function is the point where the graph of the function intersects the y-axis. This occurs when x = 0.
Step 2 ：To find the y-intercept of the function \(f(x)=\log _{3}(x+4)+2\), we substitute x = 0 into the function and solve for y.
Step 3 ：Substituting x = 0, we get \(f(0)=\log _{3}(0+4)+2\).
Step 4 ：Solving this, we get the y-intercept as approximately 3.26.
Step 5 ：None of the given options results in the correct y-intercept. The correct y-intercept for the function \(f(x)=\log _{3}(x+4)+2\) when x = 0 is approximately 3.26.
Step 6 ：\(\boxed{3.26}\) is the final answer.
