line tangent to the graph of $f(x)=(\ln x)^{5}$ at $x=8$

Step 1 ：Given the function \(f(x)=(\ln x)^{5}\), we want to find the equation of the line tangent to the graph of this function at \(x=8\).

Step 2 ：The equation of a tangent line to a function at a given point is given by \(y = f'(a)(x - a) + f(a)\), where \(f'(a)\) is the derivative of the function at the point \(a\) and \(f(a)\) is the value of the function at the point \(a\).

Step 3 ：First, we need to find the derivative of \(f(x)\). The derivative of \(f(x)=(\ln x)^{5}\) is \(f'(x) = 5(\ln x)^{4}/x\).

Step 4 ：Next, we evaluate the derivative at \(x=8\) to find \(f'(8)\). So, \(f'(8) = 5(\ln 8)^{4}/8\).

Step 5 ：We also need to evaluate \(f(x)\) at \(x=8\) to find \(f(8)\). So, \(f(8) = (\ln 8)^{5}\).

Step 6 ：Now that we have \(f'(8)\) and \(f(8)\), we can substitute these values into the equation of the tangent line to get the final equation.

Step 7 ：Substituting \(a = 8\), \(f'(8)\), and \(f(8)\) into the equation of the tangent line, we get \(y = 5*(x - 8)*\ln(8)^{4}/8 + \ln(8)^{5}\).

Step 8 ：Final Answer: The equation of the line tangent to the graph of \(f(x)=(\ln x)^{5}\) at \(x=8\) is \(\boxed{y = 5*(x - 8)*\ln(8)^{4}/8 + \ln(8)^{5}}\).