Given question is "Sketch the graph of $y=ln(2x-1)$. Determine the equation of the straight line which would need to be drawn on the same axes as graph of $y=ln(2x-1)$ in order to obtain a graphical solution of the equation $(2x-1)^2e^{x-2}=1$. Hence state the number of solutions for this equation."
We need to sketch a graph of $y=ln(2x-1)$ on solving this we get $e^y=2x-1$ . substituting this in the equation to solve we get $e^{2y}e^{x-2}=1$ which mean $y=1-\frac{x}{2}$. now I got this solution by solving analytically.
What does it mean "to determine the equation of the straight line which would need to be drawn on the same axes as graph $y=ln(2x-1)$ in order to obtain a graphical solution of the equation $(2x-1)^2e^{x-2}=1$?..." and how can we find the number of solutions for the equation?.. Kindly help me to understand the question
Graphical solution means you're solving an equation by plotting lines/curves on a graph and looking for intersection points. It's graphical because you can show where on the graph the solution is (the intersection), but you usually can't precisely read off the value.
The objective is to find the $x$ that satisfies $(2x-1)^2e^{x-2}=1$. As you have derived, the $x$ that satisfies this equation is also the $x$ that satisfies the following pair of equations $$y=\ln(2x-1) \\ y=1-\frac{x}{2}$$
If we plot the above two equations as $y-x$ curves, then the x-coordinate of the intersection points give the desired solutions for $(2x-1)^2e^{x-2}=1$. The number of solutions is just the number of intersection points. So you'll get the answer to this part of the question by plotting the above two equations on a graph.
Coming back to your doubt on the question's wording, it's saying we're going to solve $(2x-1)^2e^{x-2}=1$ by the graphical solution approach. Which means we need to plot curves on a graph. It's telling you to use $y=ln(2x-1)$ as one of the curves, and a straight line as the other. But it hasn't told you the equation of this other straight line. So that's what you need to figure out.
And you did all that; getting the equation analytically is fine, and the normal approach.