Final answer:
By taking natural logarithms and manipulating the original equation, it is shown that the equation represents a straight line when graphed with ln y against ln x. The gradient of this line is ln(3)/ln(5), and the line cuts the y-axis at the point (0, ln(3)/(2 ln(5))).
Step-by-step explanation:
The original equation is given by 5^2y = 3^(2x+1). By taking the natural logarithm of both sides we can express ln(5^2y) as 2y ln(5) and ln(3^(2x+1)) as (2x+1) ln(3), setting up the equation 2y ln(5) = (2x+1) ln(3). Dividing both sides by 2 ln(5), we get y = ((2x+1) ln(3))/(2 ln(5)), which can be expressed as y = (ln(3)/ln(5))x + (ln(3)/(2 ln(5))).
Comparing this with the general form of a straight line equation, y = mx + b, we can see that the graph of ln y against ln x is indeed a straight line. The gradient of this line is given by m = ln(3)/ln(5). The y-intercept is found when x=0 which gives us the point (0, ln(3)/(2 ln(5))) on the ln y-axis, representing the point where the line cuts the y-axis.