Logs and graphs

Hey... i'm in need of a bit of help. I have to do a Maths refresher course for my uni (I think it's like A level maths) and I'm struggling on the questions on logs. Could someone please explain to me the process that one would follow to answer either of the following questions?

1. Given that Y=5x^2, what log functions would you plot along the X and Y axes to obtain a straight line graph? Identify the gradient and the y-intercept of the straight line. There is no need to plot the graph.

2. Given that Y/4=X^3, write down this equation in the linear form y=ax+b by taking logs of both sides to base 10. What log functions would you plot along the x and the y axes to obtain a straight line graph? Identify the gradient and the y-intercept of the straight line. There is no need to plot the graph.

Reply 1

Dream Eater

Original post by ed_jnr

1. Given that Y=5x^2, what log functions would you plot along the X and Y axes to obtain a straight line graph? Identify the gradient and the y-intercept of the straight line. There is no need to plot the graph.

For the first question, take the natural logarithm of both sides to get (using standard rules of log's that you should be able to infer from the steps):

\ln y = \ln (5x^2)

\ln y = \ln 5 + \ln(x^2)

\ln y = \ln 5 + 2 \ln x

So if you plotted 'ln y' against 'ln x', you would get precisely straight line (put X = ln x and Y = ln y to see this) with y-intercept 'ln 5' and gradient 2.

EDIT: This works exactly the same for any base. You don't have to choose the natural logarithm.