Algebra Know-It-ALL

for any real number x, and

e(lny)=y

for any positive real number y.

An example

Now let’s see how a common antilog can be used to find the value of a non-whole number raised
to the power of another non-whole number. Recall the example set out earlier in this chapter:

log 10 (2.6351.078)= 1.078 log 10 2.635

≈ 1.078 × 0.421

≈ 0.454

If we get rid of all the intermediate expressions, we have

log 10 (2.6351.078)≈ 0.454

Taking the common antilog of both sides, we get

antilog 10 [log 10 (2.6351.078)]≈ antilog 10 0.454

We can use a calculator to find the common antilog of 0.454, and simplify the left side of the
equation based on the fact that the antilog function “undoes” the log function. When we do
that, we get

2.6351.078≈ 2.844

We can use the “x^y” key in a calculator to verify the above result. When I enter the original
numbers into my calculator and use that key, I get

2.6351.078≈ 2.842

This answer disagrees from the previous answer by 0.002 (or 2 parts in 1,000) because we
rounded off the calculations at every step, introducing a rounding error.
The log-antilog scheme is the way most calculators work out powers when the input
values are not whole numbers. Before the invention of logs and antilogs, expressions such as
2.6351.078 were mysterious, indeed. We can use logs and antilogs of any base to evaluate any
number raised to the power of any other number, as long as the log and the antilog are both
defined for all the arguments.

Another example

What do you get if you raise e to the power of π? You should remember from basic geometry
whatπ (the lowercase Greek letter called “pi”) means. It’s the ratio of any circle’s circumference
to its diameter, and is an irrational number equal to approximately 3.14159.

What Is an Exponential? 489