Logarithms
Logarithms, or logs are an extremely useful tool for the physicist. They were initially developed to simplify multiplication and division, and although calculators have effectively removed the need to use log tables for this, they do provide the facility to investigate the relationship between two variables, and allow us to model natural growth and decay.
The logarithm of a number is the power to which some other number, called the base, needs to be raised to in order to equal that number. During A level physics, you will come across two types of logarithms, log base ten (log) and natural logs (ln). So for example,
$$\large \log\left(100 \right )=2$$This log has a base of 10, so the base would need to be raised to the power of 2 to equal 100.
One of the most useful features of logarithms follows from the following law:
$$\large \log\left(x^{a} \right )=2\log\left(x \right )$$Therefore if two variables are related in the form of:
$$\large a=k\times b^{n}$$Where k is a constant and n is an integer. We can then use logarithms to find the value of n and k. If we ‘take logs’ on both sides of the equation we get:
$$\large \log\left(a \right )=\log\left(k \right )+n\log\left(b \right )$$This is now in the form of the equation of a straight line, $y=mx+c$. If a graph of $\log\left(a\right)$ vs $\log\left(b\right)$ then the gradient would equal the power n and the y-intercept would equal $\log\left(k\right)$.
Some common gradients and power relationships you may find are:
| Gradient | Power relationship |
|---|---|
| $1$ | $a=kb$ |
| $2$ | $a=kb^2$ |
| $-1$ | $a=\frac{k}{b}$ |
| $-2$ | $a=\frac{k}{b^{2}}$ |
| $\frac{1}{2}$ | $a=k\sqrt{b}$ |
| $-\frac{1}{2}$ | $a=-\frac{k}{\sqrt{b}}$ |
Natural Logs
Natural logarithms use the irrational number, e as the base. The base of natural logarithms, e is approximately 2.718. Natural logs are very useful when modelling natural growth or decay, as many of these systems by equations which are in the form of:
$$\large N=N_{0}e^{\lambda t}$$- Where N is the value at some time t
- N0 is the initial value of the quantity
- λ is the decay constant
- t is the time
Systems that obey this equation are said to either grow or decay exponentially, this also means that the fractional change in equal changes in time will be equal. Examples of systems that grow or decay exponentially are,
- Charging capacitors
- Nuclear decay
- Oscillator amplitude decay
- Population growth
When investigating systems which might exhibit exponential change there are two ways to progress. You can find the value of the decay constant λ by taking finding the natural log of the quantity being measured and plotting it against time.
$$\ln\left(N\right)=\ln\left(N_{0}\right)\lambda t$$Which fits the equation of a straight line where the gradient is the decay constant λ and the y-intercept would be the initial value N0.
If presented with data already plotted on a graph, you can confirm the relationship is exponential rather than $\frac{1}{x}$ or $\frac{1}{x^{2}}$ which both look quite similar. A system undergoing exponential decay, will display an equal fractional decrease, in equal intervals of time. In the diagram below of amplitude vs time, the amplitude is read at times t1, and t2, the ratio of these two amplitudes will equal the ratio of the amplitudes at times t3 and t4 if the relationship is exponential.
