Solve the following logarithmic equation. Express irrational solutions in exact form and as a rounded decimal Verify your results using a graphing utility. lnx+ln(x+8)=4 What is the exact solution? Select the correct choice below and, if necessary, fill in the answer box to complete your choice A. The solution set is (Simplify your answer. Type an exact answer. Use a comma to separate answers as needed.) B. There is no solution What is the decimal approximation of the solution?Select the correct choice below and, if necessary.fill in the answer box to complete your choice A. The solution set is (Round to three decimal places as needed. Use a comma to separate answers as needed) B. There is no solution

The problem requires us to solve a logarithmic equation and confirm our solution using a graphical approach. We need to provide both an exact form and a decimal approximation of the solution.<br /><br />Firstly, convert the given logarithmic equation from the form: <br /><br />In(x) + In(x+8) = 4<br /><br />to a single log using the property of logarithms (logb(mn) = logb(m) + logb(n)):<br /><br />In(x(x+8)) = 4 which is equal to In(x^2+8x) = 4<br /><br />It is comparing log equals to some number, we can use "change of base" formula to logarithm which converts it into:<br /><br />e^(4) = x^2+8x = 54.59815003314421<br /><br />By subtracting nature constant e^4 from either side to turn it into a quadratic formula becomes:<br /><br />x^2+8x-54.59815003314421=0<br /><br />Next, let's solve this equation. The traditional approach in solving a quadratic equation ax^2+bx+c=0 is by applying the Quadratic Formula to find the criteria for x:<br /><br />x = [-b±SQRT(b^2-4ac)]/2a<br /><br />Input a=1, b=8, c=-54.59815003314421 to the formula gives us x;<br /><br />x = [-8±SQRT(8^2-4*1*(-54.59815003314421))]/2*1<br /><br />x1 = -8+√372.6384/2 = 3.213382017<br /><br />x2 = -8-√372.6384/2 = -18.213382017<br /><br />Since a logarithm of a negative number is undefined, x can't be a negative value (-18.213382017). Therefore, only the positive solution (x=3.213382017) is physically acceptable.<br /> <br />Exact form solution: x = 3.213382017(7dp)<br /><br />Decimal approximation: The solution set is {x = 3.213} <br /><br />The solution is "The solution set {3.213}" moreover, by plotting the equation in a graphing utility you can also see that the graph{%Assistant>-- Edit here last time %} intersects with x = 3.213 only.