Chapter 6: Exponentials and logarithms

The chapter problem: the cooling coffee and the growing culture

A cup of coffee is poured at $80^\circ$C in a room kept at $20^\circ$C. The temperature above room temperature, $E = T - 20$, falls as the coffee cools:

where $E$ is in degrees Celsius and $t$ is in minutes. After $10$ minutes the coffee is at $65^\circ$C, so the excess temperature is $45^\circ$C. The coffee is ready to drink when its excess temperature is $15^\circ$C, that is, when the coffee reaches $35^\circ$C. The cooling constant $k$ is not given, and it has to be found from the one measurement.

A laboratory also grows a mould culture. The number of cells $N$ is measured each hour:

The team believes the growth follows $N = kb^t$, where $b$ is the factor by which the number multiplies each hour. They want to estimate $b$ from the data, and then predict when the number will reach $2000$.

By the end of the chapter you will answer both parts. You will find the cooling constant from the measurement, read the growth factor from the table, find the time to cool and the time to double, and connect each model to its rate of change.

First contact — no working needed yet. Look at the coffee. It cools quickly while it is hot, then more slowly as it approaches room temperature. Predict how long the coffee takes to fall from $80^\circ$C to $65^\circ$C, compared with the time to fall from $65^\circ$C to $35^\circ$C. Write both guesses down.

Then look at the table. Each number is bigger than the one before it. Work out the ratio between one count and the next, and use it to predict the count at $t = 4$ hours.

By the end of this chapter you will be able to

• sketch the graphs of $a^x$ and $e^x$, and describe their shape, asymptote and range;
• explain why the number $e$ is the natural base for an exponential model;
• differentiate $e^{kx}$ and $a^{kx}$, and interpret the result as a rate of change;
• explain why a quantity whose rate of change is proportional to its size follows an exponential model;
• convert between exponential and logarithmic form, and use the laws of logarithms;
• solve an equation of the form $a^x = b$ by taking logarithms;
• solve quadratic equations in $e^x$ or $a^x$;
• use a logarithmic graph to estimate the parameters in $y = ax^n$ and $y = kb^x$;
• model growth and decay in context, including compound interest, radioactive decay and drug concentration;
• state the limitations of an exponential model.

Where it is used

Exponentials describe anything that grows or shrinks by a fixed factor in equal steps of time. Money in a savings account, a population of bacteria, a dose of medicine, a radioactive sample and the charge in a capacitor all follow the same curve. Logarithmic scales turn that curve into a straight line, which is how scientists read a growth constant from a graph. The exponential function is also the one function whose slope equals its value, so it sits at the heart of calculus. The next chapter differentiates $e^{kx}$ and $\ln x$, and the chapter after integrates them.

Your turn to connect it

Think of something in your own life that grows or shrinks by a fixed factor. It could be a savings balance, a rumour spreading, a phone battery draining, or a cooling drink. Write down the quantity, the factor for each time step, and one question about it that a graph could answer.

Roadmap

1. 6.1 The functions a^x and e^x
2. 6.2 The gradient of e^kx and the exponential model
3. 6.3 Logarithms and the laws of logarithms
4. 6.4 Solving equations of the form a^x = b
5. 6.5 Logarithmic graphs and estimating parameters
6. 6.6 Exponential growth and decay in modelling
7. Review