# 7.3-Logarithms Advanced Level ### Question 6 #### Anna Smith (09/24/2020) --- Question 6. Compute the area of the region bounded by the $x$ axis and the curves $y=ln(x)$ and $y=ln(4-x)$. You may not use tools that we have not learned yet(for example we have not learned what the antiderivative of $ln(x)$ is). Include a graph with the appropriate region shaded. --- So we have the equations... $y=ln(x)$ and $y=ln(4-x)$ --- We need to solve this problem in terms of $y$ so the equations change to... $e^y=e^{ln(x)}$$\rightarrow$ $e^y=x$ and $e^y=e^{ln(4-x)}$ $\rightarrow$ $4-e^y=x$ is the top function and $e^y=x$ is the bottom function. --- The integral is from $0$ to $ln(2)$... $\int_0^{ln2}(4-e^y)-(e^y)$ Next take the antiderivative... The antiderivative of $4-e^y=x$ is $4y-e^y$ The antiderivative of $e^y$ is $e^y$ --- Solving this... $(4y-e^y)-(e^y)$ on the integral from $0$ to $ln2$ we get... $(4(ln2)-e^{ln2}-(4(0)-e^0))-(e^{ln2}-e^0)$ $=4ln(2)-2$ Final answer. --- <iframe src="https://www.desmos.com/calculator/nd6eci5wwi?embed" width="500px" height="500px" style="border: 1px solid #ccc" frameborder=0></iframe>