use a graphing calculator or graphing utility to sketch the graphs. label each graph. (14 points) 3.010)…

Explanation:

Step1: Recall properties of logarithmic functions

The function $y = \ln(x + 2)$ has a vertical - asymptote at $x=-2$ and passes through the point $(-1,0)$ since when $x=-1$, $y=\ln(-1 + 2)=\ln(1)=0$. The function $y = 2+\ln(x)$ has a vertical - asymptote at $x = 0$ and passes through the point $(1,2)$ since when $x = 1$, $y=2+\ln(1)=2$. Use a graphing utility to plot these two functions and label them as $y=\ln(x + 2)$ and $y=2+\ln(x)$.

Step2: Analyze exponential and logarithmic inequalities

For $y_1>4^x$, the region above the graph of $y = 4^x$ is considered. For $y_2\leq3+\log(x)$ (assuming base - 10 if not specified otherwise), the region below or on the graph of $y = 3+\log(x)$ is considered. The vertical asymptote of $y = 3+\log(x)$ is $x = 0$. Use a graphing utility to plot $y = 4^x$ and $y=3+\log(x)$. Then shade the region that satisfies both $y>4^x$ and $y\leq3+\log(x)$.

Answer:

The graphs of $f(x)=\ln(x + 2)$ and $g(x)=2+\ln(x)$ are to be plotted using a graphing utility and labeled. For the second part, the graphs of $y = 4^x$ and $y=3+\log(x)$ are plotted using a graphing utility, and the region that satisfies $f(x)>4^x$ and $g(x)\leq3+\log(x)$ is shaded. The actual sketches and shadings are to be done on the provided grids with a graphing calculator or utility.