Logarithmic functions

Exercise 1

The population of a medium size town has been growing at a rate of (about) $6$ % per year since 1-1-2000. There were $80000$ inhabitants on 1-1-2000.

a

Set up a formula to calculate the number of inhabitants $A$ as a function of time $t$ in years since 1-1-2000.

b

Draw the corresponding graph on semi-logarithmic paper.

c

On this graph, find the number of inhabitants on 1-1-2015. Check your answer using the formula.

Exercise 2

The table below shows measurements made in a bacterial culture. $t$ is given in hours, and $N (t)$ is the number of bacteria.

t	0	1	2	3	4	5	6
N	50	84	141	237	398	670	1125

a

Use these measurements to make another table showing $log (N)$ against $t$ .

b

Draw the corresponding graph. Can you approximate the graph with a straight line? Is this exponential growth?

c

Write down a formula of $log (N)$ as a function of $t$ .

d

Using your answer in c, write down a formula for $N$ as a function of $t$ .

Exercise 3

The graph of an increasing amount $V$ as a function of time $t$ is shown here plotted on semi-logarithmic paper.

a

Set up a formula for $V (t)$ .

b

Calculate, rounded to two decimals, the value of $t$ where $V (t) = 10$ . Use the graph to check your answer.

c

Somewhere in the negative values of $t$ , the graph has an intersect with the $t$ -axis. Calculate the corresponding value of $t$ , rounded to two decimals.

Exercise 4

Mammals change from trot to gallop when they reach a certain threshold pace (the number of steps per minute). The pace at which this happens appears to depend on the body weight (in kg). Take $m$ (in kg) to be the body weight and $P$ to be the pace. The straight line goes through the points belonging to small dogs and horses.

a

How can you see that a logarithmic scale has been used on both axes?

b

Since a logarithmic scale has been used on both axes, the plot actually shows the relationship of $log (P)$ with $log (m)$ . The point for horses then represents $log (m) = 2.9$ and $log (P) = 2.0$ . Determine yourself the (approximate) values for the point belonging to the small dog.

c

Now derive a formula for $log (P)$ as a function of $log (m)$ .

d

Using the properties of logarithms you can now derive a formula for $P$ as a function of $m$ . Show all steps in your derivation.

Exercises