Logarithms and Exponentials
Logarithms, Laws, Natural Logarithms, Exponential Growth and Decay, Reduction to Linear Form
Why we study this
We used to use logarithms as an aid to calculation, either using tables of logarithms or as the principle behind slide rules. Logs allow us to use addition to multiply, use subtraction to divide, multiplication for powers and division for roots. However, logarithms have a more fundamental role in mathematics as the inverse function to the exponential or power function, and as such has applications for the modelling of growth and decay, for example radioactivity, cooling of liquids, and modelling population growth.
History of Logarithms
The story of logarithms begins in a drafty Scottish castle during the late 16th century. John Napier, the 8th Laird of Merchiston, was a man of many interests, from theology to agriculture, but he became obsessed with solving a specific problem: the sheer exhaustion of manual arithmetic. For astronomers and sailors of his era, a single calculation involving large numbers could take days and was prone to human error. Napier spent over twenty years searching for a shortcut. In 1614, he finally published his discovery, introducing a system that effectively allowed users to bypass tedious multiplication. His work was so revolutionary that Johannes Kepler, the famous astronomer, used these methods to decode the movements of the planets. Napier also pioneered the use of the decimal point, further simplifying how we look at numbers. For nearly four centuries after his death, his invention remained the primary tool for high-level computation. If you had been a student before the 1970s, you wouldn't have reached for a digital device; instead, you would have mastered the sliding scales of a slide rule or flipped through pages of printed tables, all carrying the legacy of Napier’s quest to make the difficult simple. His genius transformed the speed of human discovery long before electricity existed.
Logarithms Exam Tips
The crucial step to understanding logarithms is the relationships between them and exponents or indices. Once this is understood, the laws of logarithms effectively become a re-formulation of the laws of indices. It is therefore important to know your index laws very thoroughly, and be clear about how they lead to the laws of logarithms.
Definition of logarithms
First off, it is important to be able to find logarithms of numbers to any base without using a calculator, but simply from the definition. For example, the log to base 2 of 8 is 3, the log to base 10 of 0.1 is -1, and the log to base e of e^5 is 5. Here's a reminder of how logs are defined, and some practice to sharpen up your logging. [text]
Laws of logarithms
A lot of exam questions rely on manipulation expressions with logarithms using the laws they obey. The relationship which these laws have with the laws of indices is entirely explained by the fact that finding a logarithm of x to base a is the inverse process to raising a to the power x. Watch a video [text]
Natural Logarithms
The number e first appeared in the 17th century when mathematicians studied compound interest and continuous growth. They discovered that as interest is compounded more and more frequently, the growth approaches a limiting value, now called e, Here is a brief video on this [text]. approximately 2.718. Natural logarithms are logarithms with base e, and they became central in calculus because they simplify formulas involving rates of change, exponential growth and decay, and many real-world processes in science and engineering.
Exponential graphs
The graph y = a^x, where a is positive, is an exponential graph. For a > 1, we have exponential growth; when 0 < a < 1 we have exponential decay. It can be shown that the rate of change of any exponential function is proportional to its value, which makes it a good basic model for population growth, which grows in proportion to its size, or radioactive decay, which declines on proportion to its mass.
The constant of proportionality turns out to be ln a. When a = e, this constant is ln e or 1. So the rate of change, or derivative of e^x is just e^x! This explains one of the important properties of the number e.
The graph of y = e^x can also be thought of as the limit of the graph of y = (1 + 1/n)^(nx), as shown in the video below. As e is approached by (1 + 1/n)^n, g(x) = e^x is approached by
f(x) = [(1+1/n)^n]^x = (1 + 1/n)^(nx).
Exponential Growth and Decay
Many A level questions apply logs and exponentials to practical problems involving growth and decay. Hints: 'initial' means when t = 0; 'long term' means t is infinite so e⁻ᵏᵗ will be zero.
Try this example. Get the solution, click the video below.
The temperature T in °C of a cup of water after t minutes is modelled by the equation T = a + be⁻ᵏᵗ, where k is a positive constant. Initially the temperature is 100 °C and room temperature is 21 °C.
(a) Find a and b.
After 1 minute, the temperature has gone down to 93 °C.
(b) Find how long it takes to drop to 50 °C.
Reduction to linear form
A common application of logarithms is to confirm a relationship between variables by plotting the logs of one or both of the variables and finding a linear relationship.
Suppose y varies exponentially with x, or y = be-kx. In this case plotting the logarithms of the y-values against x would result in a straight line.
On the other hand, if y is proportional to a power of x, or y = a xᵇ, then plotting the logs of the y-values against the logs of the x values produces a straight line. Here is why. [text]
Kepler's Third Law: how logarithms reveal the motion of the planets
Example: Kepler's third law states that the square of a planet's orbital period, T, is proportional to the cube of its orbital distance from the Sun, R:
Here is the data for the planets' orbital period (in years) and orbital diatance (in Astronomical units - 1 AU is the distance from the Earth to the Sun:
The graph of T against R seems to follow a curve, but what curve? Well, if we plot ln R against ln T, we get a more obvious relationship:
What's the equation of this perfect straight line?
Well, the intercept is at the origin and, looking at the point for Saturn, the gradient is 1.5/1 or 1.5
So ln R = 0 + 1.5 ln T
and e^(ln R) = e^(1.5ln T) = (e^lnT)^1.5
or R = T^1.5, and R^2 = T^3, as Kepler's Third Law states!