doPhysics · Mathematical Tools · Toolbox

Exponentials and Logarithms

Why this chapter

Why this chapter exists

You may go on with the physics now. When a lesson needs this tool, it will tell you and bring you here.

A capacitor empties through a resistor, a nucleus decays, a cup of chai cools, a swing slowly comes to rest. Each of them falls by the same fraction in every equal interval of time, and each follows one curve, the exponential. Its inverse, the logarithm, dates a rock, measures a sound in decibels and turns a bending graph into a straight line. This chapter teaches the two functions as the physics uses them, with the physics chapters' own numbers.

You will learn

By the end of this chapter you will

Recognise change by a constant factor, and use \(y_0r^n\) and the sum of a geometric series.
Say why the number \(e\) appears, and use \(d(e^x)/dx = e^x\).
Use the time constant and the half-life of \(y_0e^{-t/\tau}\), and read \(\tau\) from the tangent at the start.
Use the approach \(y_f(1 - e^{-t/\tau})\) for capacitors, coils and terminal velocity.
Solve for a time in the power with logarithms, and use their rules and the change of base.
Work with the decibel scale, and read a decay constant or a power from a log graph.

Before you begin

You should be able to use powers such as \(10^3\) and \(2^{-1}\), and read a value from a graph. The first lessons use the idea of a slope and a rate of change; the chapter Differentiation explains them fully. A calculator with \(e^x\), \(\ln\) and \(\log\) keys helps, but the values you need are given.

Already know this?

Lessons

Growth and Decay by a Constant Factor

A bouncing ball, a chain reaction and a decaying nucleus all change by the same factor in every equal step, \(y_n = y_0 r^n\). Making the steps smaller and smaller leads to the number \(e = 2.718\), the one base whose curve has a slope equal to its height.

Growth and decay by a constant factorE is the base whose slope equals its heightA rate proportional to the amount

15 min0%

→

The Decay Curve, the Time Constant and the Half-Life

Anything that falls at a rate proportional to itself follows \(y = y_0e^{-t/\tau}\). The time constant \(\tau\), the half-life \(\tau\ln 2\), and the tangent at the start that meets the time axis at \(\tau\), for capacitors, nuclei, cooling tea and dying swings alike.

Time constant and half lifeThe half life halves from any startThe tangent at the start meets the axis at tauOne curve many time constants

15 min0%

→

The Exponential Approach to a Final Value

A capacitor charging, a current growing in a coil and a ball speeding up in oil all climb as \(y = y_f(1 - e^{-t/\tau})\). The gap to the final value decays exponentially, so the climb reaches 63 % at one time constant and never quite ends. Four exponential curves, side by side.

Exponential approach to a final valueThe gap to the final value decaysFour exponential curves side by side

14 min0%

→

The Logarithm Undoes the Exponential

When the unknown sits in the power, as the age of a rock does in \(e^{-\lambda t}\), the logarithm frees it. What a logarithm is, the three rules that turn products into sums, the natural and the common logarithm, and the change of base.

The logarithm undoes the exponentialRules of logarithmsNatural and common logarithms

15 min0%

→

Logarithmic Scales and the Decibel

The ear hears a whisper and a jet engine, whose intensities differ a hundred thousand million times. A logarithmic scale gives equal space to equal ratios, so the whole range fits between 0 and 140 dB. Why ratios become differences, why doubling adds 3 dB, and why levels never simply add.

Logarithmic scales and the decibelRatios of intensity become differences of level

14 min0%

→

Straight Lines on Log Graphs

A decay curve becomes a straight line when \(\ln N\) is plotted against time, and a power law becomes one on log-log axes. Reading a decay constant, a cooling constant or the power of a law from the slope, and knowing when the line is honest.

Straight lines on log graphsReading a decay constant from a semi log graphLog log graph of a power law

14 min0%

→

Read in order. The small labels are the topics of each lesson. They take your colour as you answer the checks.

Glossary

The key words of this chapter, with the meaning this chapter gives them.

common logarithm: The logarithm to the base 10, written \(\log\). \(\ln x = 2.303\log x\).
decibel: The unit of sound level, a tenth of a bel. The level \(\beta = (10\,\text{dB})\log_{10}(I/I_0)\) compares an intensity with \(I_0 = 10^{-12}\,\text{W/m}^2\).
e: The number \(2.71828\ldots\), the limit of \((1 + 1/n)^n\). It is the one base whose exponential curve has a slope equal to its height.
exponential: A change in which the number of steps, or the time, sits in the power, as in \(y_0r^n\) or \(y_0e^{kt}\). Equal steps give equal ratios.
geometric series: A sum of terms each of which is a fixed factor \(r\) times the one before. For \(0 < r < 1\) the endless sum is \(a/(1 - r)\), where \(a\) is the first term.
half-life: The time in which an exponentially decaying quantity falls to half of any value, \(T_{1/2} = \tau\ln 2 = 0.693\tau\).
log-log graph: A graph with logarithms on both axes. A power law is a straight line on it, and the slope is the power.
logarithm: The power to which a base must be raised to give a number: \(\log_b y = x\) when \(b^x = y\). It exists only for positive numbers.
logarithmic scale: A scale on which each number is marked at a distance proportional to its logarithm, so that equal ratios take equal lengths.
natural logarithm: The logarithm to the base \(e\), written \(\ln\). It undoes the exponential: \(\ln e^x = x\).
power law: A law of the form \(y = kx^n\), in which one quantity is a fixed power of another.
semi-log graph: A graph with a logarithm on one axis and an ordinary scale on the other. An exponential law is a straight line on it.
time constant: The time \(\tau\) in which a quantity decaying as \(e^{-t/\tau}\) falls to \(1/e\), about 37 %, of any value. For a capacitor \(\tau = RC\), for a coil \(\tau = L/R\).