Using logarithmic scales on graphs Problem set B

The masses of four animals are 0.06 g, 90 g, 7500 g and 4 800 000 g.

1. (a) Write the log10 of each value, to 3 significant figures.
2. (b) By how much does the log10 of a mass change when the mass is divided by 100?

The value y (in euros) of an investment after x years follows \(y=a\times b^{x}\). The graph of \(\log_{10}y\) against x is a straight line through (0, 1.4) and (6, 2.0).

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 = 15.

The surface area y of a sphere-shaped cell follows \(y=a\,x^{k}\) in terms of its volume x. The graph of \(\log_{10}y\) against \(\log_{10}x\) is a straight line through (0, 0.5) and (3, 2.3).

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 = 200.

The mass y (in grams) of a growing crystal after x days 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 activity N (counts per minute) of a radioactive sample is measured every day. 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 = 15.

The time T (s) for a pendulum to swing back and forth is measured for five lengths L (m). A model \(T=a\,L^{k}\) is suggested.

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

An engineer 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 × b^x.

Two relationships are \(y=40\times0.65^{x}\) and \(y=8\,x^{2}\).

1. (a) Write log10 y as a linear expression in x for \(y=40\times0.65^{x}\).
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 \(y=8\,x^{2}\), and state its gradient and intercept.