Algebra Know-It-ALL

We can raise negative numbers to real-number powers, but this is rarely done with loga-

rithmic functions. We aren’t likely to encounter any “base-(−10)” or “base-(−e)” logarithms,

for example, although technically there is no reason why such things can’t exist.

Common logs

Base-10 logarithms are also known as common logarithms or common logs. In equations, com-

mon logs are denoted by writing “log” with a subscript 10, and then the number, called the

argument, for which you want to find the logarithm. Here are a few examples that you can

verify with your calculator:

log 10 100 = 2

log 10 45 ≈ 1.653

log 10 10 = 1

log 10 6 ≈ 0.7782

log 10 1 = 0

log 10 0.5 ≈−0.3010

log 10 0.1 =− 1

log 10 0.07 ≈−1.155

log 10 0.01 =− 2

The squiggly equals sign means “is approximately equal to.” You’ll often see it in scientific and

engineering papers, articles, and books. The first equation above is another way of writing

102 = 100

You could also say “The common log of 100 is equal to 2.” The second equation is another

way of writing

10 1.653≈ 45

You could also say “The common log of 45 is approximately equal to 1.653.”

Natural logs

Base-e logarithms are also called natural logs or Napierian logs. In equations, the natural-log

function is usually denoted by writing “ln” or “loge” followed by the argument. Here are some

equations using the natural-log function, which you can check out with your calculator:

ln 100 ≈ 4.605

ln 45 ≈ 3.807

ln 10 ≈ 2.303

ln 6 ≈ 1.792

ln e= 1

ln 1 = 0

480 Logarithms and Exponentials