ln (1/e)=− 1ln 0.5 ≈−0.6931
ln 0.1 ≈−2.303
ln 0.07 ≈−2.659
ln 0.01 ≈−4.605
The first equation above is another way of writing
e4.605≈ 100You would say it as “The natural log of 100 is equal to approximately 4.605.” The second
equation is an alternative way writing
e3.807≈ 45You could also say “The natural log of 45 is approximately equal to 3.807.”
The arguments in common or natural log functions don’t have to be whole numbers,
fractions, or terminating decimals. You can have logs of arguments that are irrational, such as
π,e, the positive square root of 2, or the cube root of 100.
Don’t let them confuse you!
Authors don’t all agree on what the notation “log” means. In some texts, natural logs (that is, base-e logs)
are denoted by writing “log” without a subscript, followed by the argument. But in other texts and in most
calculators, “log” means the common (base-10) log.
To avoid confusion, always include the base as a subscript whenever you write “log” followed by an
argument. For example, write “log 10 ” or “loge” instead of “log” all by itself. You don’t have to use a subscript
when you write “ln” for the natural log.
If you aren’t sure what the “log” key on a calculator does, do a trial calculation to find out. If the “log”
of 10 equals 1, then it’s the common log. If the “log” of 10 equals an irrational number slightly larger than
2.3, then it’s the natural log.
Here’s a challenge!
Compare the common logarithms of 0.01, 0.1, 1, 10, and 100. Then compare the natural logarithms of
those same arguments.
Solution
Note that 0.01 = 10 −^2 , 0.1 = 10 −^1 , 1 = 100 , 10 = 101 , and 100 = 102. Therefore:
log 10 0.01 =− 2log 10 0.1 =− 1
log 10 1 = 0
log 10 10 = 1
log 10 100 = 2
What Is a Logarithm? 481