Exponents and Logarithms: A World That Grows by Multiplying

1.Powers and the Rules of Exponents

Multiplying the same number by itself several times is called raising it to a power, and the number of times it is multiplied is called the exponent. 2⁵ = 2 × 2 × 2 × 2 × 2 = 32; here 2 is the base and 5 is the exponent. Multiplying 2 together 10 times gives 1,024, more than 500 times where you started. Compare that with adding 2 together 10 times, which gives 20, and you can see how fast repeated multiplication grows.

The rules of exponents follow directly from the idea of "counting how many times you multiply." 2³ × 2⁴ multiplies 2 together 3 times and then 4 more times, so 2 is multiplied 7 times in all: 2⁷. For division you subtract the counts, and for a power of a power you multiply them.

2.Compound Interest: Interest on Interest

With simple interest, interest is paid only on the original principal; with compound interest, the interest is added to the principal and the next interest is paid on that total. After n years at a compound rate of r per year, the principal has been multiplied by (1 + r) n times, so compound interest is a question of exponents.

In the first few years, the gap between simple and compound interest looks small, but the longer the period, the faster the gap widens. With loans, the same principle works against you: when unpaid interest itself earns interest, debt also grows exponentially.

3.Doubling Time and the Rule of 72

The time it takes a compounding quantity to double can be estimated as "72 ÷ interest rate (%)." This is called the rule of 72. At 6% a year, 72 ÷ 6 = 12, so about 12 years; at 8% a year, about 9 years.

It is only an estimate. Compounding at 6% a year for 12 years gives 1.06¹² ≈ 2.012, which is nearly exact, but when the rate is very high or very low, the error grows. The exact period is found with logarithms, covered in the next section. For 6% a year, the exact value is about 11.9 years.

The rule works for more than money. It applies to anything that grows by a fixed percentage each year: number of users, prices, amounts of data, and so on. If prices rise 3% every year, they double in about 72 ÷ 3 = 24 years, and the amount the same money can buy is cut in half.

4.Logarithms: How Many Times Was It Multiplied?

How many times do you multiply 2 to get 8? 3 times. The word for this "how many times" is logarithm. You write log₂ 8 = 3 and read it as "log base 2 of 8 is 3." It is the reverse question of the exponent statement 2³ = 8. log₁₀ 1000 = 3 and log₂ 1024 = 10 mean the same kind of thing.

The most important property of logarithms is that they turn multiplication into addition, because the counts of multiplications add up: log(a × b) = log a + log b. That is why the base-10 logarithm goes up by 1 every time you multiply by 10.

Logarithms also answer "how many times do you have to cut it in half?" When you search for a value among 1,000,000 sorted items by looking at the middle and throwing away half each time, log₂ 1,000,000 ≈ 19.9, so you find it in about 20 steps at most. That's because 2²⁰ = 1,048,576 is bigger than 1,000,000. This is one of the principles that let computers search large amounts of data quickly.

5.Reading Log-Scale Graphs

On an ordinary scale, each step adds the same amount (0, 10, 20, 30, …). On a log scale, each step multiplies by the same factor (1, 10, 100, 1000, …). That is why a quantity growing exponentially becomes a straight line when plotted on a log scale. A quantity that doubles every year is a curve that suddenly shoots up on an ordinary scale, but a line with a constant slope on a log scale.

When reading a log-scale graph, remember that equal differences in height mean equal multiples. The distance from 10 to 100 is the same as the distance from 100 to 1000. If you read it like an ordinary scale, you will underestimate the enormous growth toward the end. Sound intensity (decibels) and earthquake magnitude are also logarithmic scales, so a difference of just 1 or 10 in the number can mean the actual intensity differs many times over.

Conversely, a log scale is useful when you want to see whether growth is slowing down. If the slope of a log graph is decreasing, the growth rate is falling.

• If the axis is marked in multiples, like 1, 10, 100, 1000, it is a log scale
• A straight line on a log scale = growth at a constant rate (exponential growth)
• Equal differences in height = equal multiples, not equal amounts