Considerπ equal to 3.14159, and consider e equal to 2.71828. Then
log 10 πe≈ 2.71828 log 10 3.14159
≈ 2.71828 × 0.49715
≈ 1.35139
When you take the common antilog of this, you’ll get πe because
antilog 10 (log 10 πe)=πe
Calculating, you should get
antilog 10 1.35139 ≈ 22.45898
When I input the numbers into my calculator and use the “x^y” key, I get
3.141592.71828≈ 22.45906
Rounding error strikes again! And as before, if you wish, you can see for yourself how this error shrinks to
the vanishing point if you let your calculator keep all of its extra digits until the final step.
How Exponentials Work
In most real-life applications of exponentials, the base b is either 10 or e. However, once in a
while you’ll come across a situation where b is some other positive real number. This section
describes the basic properties that hold for exponentials in general.
Reciprocal vs. negative exponent
Suppose that x is some real number. The reciprocal of the exponential of x is equal to the
exponential of the negative of x, as follows:
1 / (bx)=b−x
whenb > 0. You should recognize this from your work with powers and roots. Here’s a famil-
iar example. You know that 1/8 is equal to 1 / (2^3 ). This is the same as saying that 1/8 is equal
to 2−^3. You also know that 1/100 equals 1 / (10^2 ), which is the same as saying that 1/100
equals 10−^2. Now consider this, rounded to four decimal places:
1 / (e^3 )≈ 1 / (2.718^3 )
≈ 1 / 20.079
≈ 0.0498
How Exponentials Work 491
