Using logarithmic scales on graphs Problem set A

The areas of four lakes are 4 km², 250 km², 18 000 km² and 82 000 km².

1. (a) Write the log10 of each value, to 3 significant figures.
2. (b) By how much does the log10 of an area increase when the area is multiplied by 10?

A rumour spreads through a school. The number of people y who have heard it after x hours follows \(y=a\times b^{x}\). The graph of \(\log_{10}y\) against x is a straight line through (0, 0.9) and (5, 2.4).

1. (a) Write down the gradient and the y-intercept of the line.
2. (b) Find the values of a and b.
3. (c) Find y when x = 9.

The strength y of a signal at distance x from a mast follows \(y=a\,x^{k}\). The graph of \(\log_{10}y\) against \(\log_{10}x\) is a straight line through (0, 1.3) and (2, 3.1).

1. (a) Write down the gradient and the y-intercept of the line.
2. (b) Find the values of a and k.
3. (c) Find y when x = 50.

The number of people y who follow a page after x weeks is recorded. Test the model \(y=a\times b^{x}\).

1. (a) Find log10 y for each value.
2. (b) Show that the points of log y against x lie on a straight line.
3. (c) Write the model, in the form a × b^x.

The number of yeast cells N (thousands) in a sample is counted every hour. A model \(N=a\times b^{t}\) is suggested.

1. (a) Use your calculator to fit a straight line to log10 N against t. Write down the gradient, the intercept and r.
2. (b) Write the model in the form N = a × b^t.
3. (c) Use the model to estimate N when t = 10.

The power output P (kW) of a wind turbine is measured at five wind speeds v (m/s). A model \(P=a\,v^{k}\) is suggested.

1. (a) Use your calculator to fit a straight line to log10 P against log10 v. Write down the gradient, the intercept and r.
2. (b) Write the model in the form P = a × v^k.
3. (c) Use the model to estimate P when v = 14.

A scientist records y for six values of x and must choose between an exponential model and a power model.

1. (a) Find the correlation coefficient r for log10 y against x.
2. (b) Find the correlation coefficient r for log10 y against log10 x.
3. (c) State which model fits best and write it in the form y = a × x^k.

Two relationships are \(y=12\times0.8^{x}\) and \(y=5\,x^{3}\).

1. (a) Write log10 y as a linear expression in x for \(y=12\times0.8^{x}\).
2. (b) State the gradient and the intercept of that line, and say why the gradient is negative.
3. (c) Write log10 y as a linear expression in log10 x for \(y=5\,x^{3}\), and state its gradient and intercept.