Mathematics provides a means of tracking, comparing and expressing data that vary broadly in scale.
When COVID-19 hit the United States, the numbers just seemed to explode. First, there were only one or two cases. Then there were 10. Then 100. Then thousands and then hundreds of thousands. Increases like this are hard to understand. But exponents and logarithms can help make sense of those dramatic increases.
Scientists often describe trends that increase very dramatically as being exponential. It means that things don’t increase (or decrease) at a steady pace or rate. It means the rate changes at some increasing pace.
An example is the decibel scale, which measures sound pressure level. It is one way to describe the strength of a sound wave. It’s not quite the same thing as loudness, in terms of human hearing, but it’s close. For every 10 decibel increase, the sound pressure increases 10 times. So a 20 decibel sound has not twice the sound pressure of 10 decibels, but 10 times that level. And the sound pressure level of a 50 decibel noise is 10,000 times greater than a 10-decibel whisper (because you’ve multiplied 10 x 10 x 10 x 10).
An exponent is a number that tells you how many times to multiply some base number by itself. In that example above, the base is 10. So using exponents, you could say that 50 decibels is 104 times as loud as 10 decibels. Exponents are shown as a superscript — a little number to the upper right of the base number. And that little 4 means you’re to multiply 10 times itself four times. Again, it’s 10 x 10 x 10 x 10 (or 10,000).
Logarithms are the inverse of exponents. A logarithm (or log) is the mathematical expression used to answer the question: How many times must one “base” number be multiplied by itself to get some other particular number?
For instance, how many times must a base of 10 be multiplied by itself to get 1,000? The answer is 3 (1,000 = 10 × 10 × 10). So the logarithm base 10 of 1,000 is 3. It’s written using a subscript (small number) to the lower right of the base number. So the statement would be log10(1,000) = 3.
At first, the idea of a logarithm might seem unfamiliar. But you probably already think logarithmically about numbers. You just don’t realize it.
Let’s think about how many digits a number has. The number 100 is 10 times as big as the number 10, but it only has one more digit. The number 1,000,000 is 100,000 times as big as 10, but it only has five more digits. The number of digits a number has grows logarithmically. And thinking about numbers also shows why logarithms can be useful for displaying data. Can you imagine if every time you wrote the number 1,000,000 you had to write down a million tally marks? You’d be there all week! But the “place value system” we use allows us to write down numbers in a much more efficient way.
