Exponents and Logarithms

Functions and Graphs ended with a falling stone whose graph bent upward because each second added more distance than the one before. That growth was still addition, only faster and faster addition. Multiplying is a different animal. A story told to every first-year engineering student: take a job that pays one penny on day one, doubled every day for a month. By day 30 the single day's pay is 5,368,709 dollars, and the month's total lands at 10,737,418.23 dollars. No addition-based rule in the last lesson produces numbers like that from a starting point of one cent. Multiplying by the same factor every step does, and that pattern has a name, a notation, and an inverse, all covered here.

Powers are repeated multiplication

An exponent records repeated multiplication. In 2^5, the base 2 is multiplied by itself 5 times: 2 times 2 times 2 times 2 times 2 = 32. The exponent counts the multiplications, not the factors, which is why 2^1 is just 2. The general form b^n means n copies of b multiplied together.

The familiar rules are not arbitrary; each one falls out of what repeated multiplication means. Multiply two powers of the same base and the exponents add, because the factors pile up in one chain: 2^3 times 2^4 is (2 times 2 times 2) times (2 times 2 times 2 times 2), which is 2^7. Divide powers of the same base and the exponents subtract, because factors cancel in pairs. Raise a power to a power and the exponents multiply: (2^3)^2 is 2^3 times 2^3, which is 2^6. Zero and negative exponents follow from the subtraction rule rather than from intuition. Since 2^3 divided by 2^3 is 1, and the rule says the exponents subtract to give 2^0, it must be that 2^0 = 1. Any nonzero base to the zero power is 1. A negative exponent means division: 2^-3 = 1/(2^3) = 1/8. Nothing here needs memorizing as a list; every rule is the subtraction rule in disguise.

Scientific notation: engineering's shorthand

The quantities engineers handle span absurd ranges. The mass of an electron is about 9.1 times 10^-31 kilograms. The distance from Earth to the sun is about 1.5 times 10^11 meters. The number of molecules in a mole of gas is about 6.0 times 10^23. Writing such numbers out digit by digit buries the useful part, a couple of significant figures, under a parade of placeholder zeros, and every extra zero is one more chance for a copy error.

Scientific notation writes a number as a value between 1 and 10 multiplied by a power of ten. The speed of light is about 3.0 times 10^8 meters per second; electron mass is 9.1 times 10^-31 kilograms. The form makes the size obvious at a glance: the exponent is a rough order of magnitude, and the leading digits are the precision. Multiplying is two easy operations: multiply the leading values, add the exponents. (2 times 10^3) times (4 times 10^5) is 8 times 10^8. Dividing subtracts the exponents the same way.

Python prints these numbers in e-notation, where 9.1e-31 reads as 9.1 times 10 to the -31:

electron = 9.1e-31        # mass of an electron, kilograms
earth = 5.97e24           # mass of the Earth, kilograms
print(earth / electron)   # electrons needed to equal Earth's mass
print(2e3 * 4e5)          # leading values multiply, exponents add

The first line prints about 6.56e54, meaning roughly 6.6 times 10^54 electrons would weigh as much as the Earth. The second prints 800000000.0, which is 8 times 10^8. The e-notation is not a different system, just Python's spelling of the same one.

The notation also turns awkward divisions into quick ones. Light takes about 500 seconds to reach Earth from the sun: 1.5 times 10^11 meters divided by 3.0 times 10^8 meters per second gives 0.5 times 10^3, which is 500 seconds, a little over 8 minutes. The leading values divided to 0.5, the exponents subtracted to 3, and the answer carried the right units through the whole operation. That is the notation doing its job: keeping the arithmetic small enough to do mentally while the units keep it honest.

Exponential growth: multiplication as the rate

Exponential growth is the pattern "multiply by the same factor every step." Written as a function, N = N0 times (growth factor)^t, where N0 is the starting amount and t counts the steps. With a doubling pattern the factor is 2, and the time a quantity needs to double is called the doubling time. Money at 7 percent annual growth doubles in about a decade; the handy rule of 70 says the doubling time in years is roughly 70 divided by the growth rate in percent.

The e in the exponent rules is worth meeting properly. The number e, about 2.71828, is the natural base for growth that compounds continuously, with no pauses between steps. Continuous growth at rate r follows A = P times e^(rt): 1000 dollars at 5 percent for 10 years gives 1000 times e^0.5, about 1648.72. Discrete compounding and continuous compounding land within a few percent of each other over human timescales, but the continuous form shows up everywhere in engineering because so many physical processes, radioactive decay and capacitor charging among them, flow continuously rather than in yearly ticks.

The reason exponential growth feels violent is that each doubling doubles the size of every future doubling. Ten doublings make about a thousand: 2^10 = 1024. Twenty doublings make about a million. Thirty make about a billion. The penny story again, this time computed rather than promised:

pay = 1        # pennies earned on day 1
total = 0
for day in range(1, 31):
    total += pay
    if day < 30:
        pay *= 2

print(pay / 100)     # day 30 pay alone, in dollars
print(total / 100)   # the whole month, in dollars

Day 30 pays 5,368,709.12 dollars and the month totals 10,737,418.23. Push the same deal to 40 days and the total crosses 10 billion dollars, because 2^40 is 1,099,511,627,776 and even pennies multiply into serious money. Nothing about the deal changed except the number of doublings, which is the whole character of exponential growth: the curve stays boring for a shockingly long time, then goes vertical.

Half-life: decay runs the same rule backward

Exponential decay is the same pattern with a factor below 1. The standard measure is half-life, the time a decaying quantity takes to fall to half its value. Carbon-14, the isotope used to date organic remains, has a half-life of 5730 years. Start with 100 grams: 50 grams remain after 5730 years, 25 after 11,460, 12.5 after 17,190, and 6.25 after 22,920. Each half-life knocks the amount down by the same factor, so the absolute loss shrinks every round: the curve falls fast at first and then flattens, the mirror image of the penny curve's climb.

Halving is multiplying by 1/2, which the exponent rules from the top of the lesson write as 2^-1. That is why decay formulas are full of negative exponents: N = N0 times (1/2)^(t/5730) and N = N0 times 2^(-t/5730) are the same sentence in two dialects. The curve never quite touches zero, which is why carbon-14 dating still finds a signal tens of thousands of years later, though past about eight half-lives the remainder is small enough that measurement noise takes over.

Archaeology runs the clock with this rule. A bone found holding one eighth of the carbon-14 that living tissue carries has stood through 3 halvings, and 3 times 5730 is 17,190 years, so the animal died roughly 17,000 years ago. One measurement, one small multiplication, and a date falls out. The halvings also explain the honest limit of the method: past about 50,000 years too little carbon-14 remains for the ratio to be read with any confidence, and other isotopes with longer half-lives take over the job.

The logarithm runs multiplication backward

A logarithm answers one question: what exponent produced this? The statement 2^3 = 8, read backward, is log2(8) = 3. The logarithm is the inverse of the power: powers build 2, 4, 8, 16, and logarithms run the ladder down, asking at which rung a given number sits. log10(1000) = 3 because 10^3 = 1000, and the natural logarithm, written ln, is the same question with base e.

Log scales exist because some ranges are too wide for any ruler. Sound, acidity, and earthquake size all span factors of millions or billions, so engineers mark the scale in exponents instead. The pH scale measures acidity as the negative logarithm of hydrogen ion concentration in moles per liter: a pH 3 solution sits at 0.001 mol/L and a pH 5 solution at 0.00001 mol/L, so pH 3 is 100 times more acidic than pH 5. The Richter scale works the same way on earthquake amplitude, one whole number meaning a factor of 10. Decibel arithmetic gives a factor of 10 in power for every 10 dB, so a 130 dB jet engine pushes about 10 million times the acoustic power of a 60 dB conversation. On each of these scales, small-looking steps are enormous multiples, which is exactly the point.

import math

print(math.log2(1024))        # the exponent that makes 2 produce 1024
print(-math.log10(1e-3))      # pH of a solution at 0.001 mol/L
print(10 * math.log10(1e7))   # decibels for a 10-million-fold power ratio

The outputs are 10.0, 3.0, and 70.0. Each line is a real conversion a working engineer might need: a doubling count, a pH, a decibel level, and all three are just "solve b^x = n for x."

Every tool in this lesson handles size: how large, how fast growing, how compressed a scale has to be to fit on one axis. None of it can say which way anything points. Trigonometry and Vectors adds direction, and direction turns out to need triangles.