Algebra Know-It-ALL

If we let the calculator keep all its extra digits during the calculation process, we don’t get the

rounding error when we go back to three decimal places at the end. (Try it and see.) Now let’s

useb= 10 instead of b=e. In that case, the left side becomes

(10^2 )^3 = 1003

= 1,000,000

= 106

and the right side becomes

10 (2×3)= 106

There’s no rounding error here, because the values are exact throughout!

Mixing exponentials in a product

We can express the product of a common exponential and a natural exponential a “mutant

exponential” whose base is 10 times e. If x is the argument of both exponentials, then

(10x)(ex)= (10e)x

This is an adaptation of the power of product rule from Chap. 9. Let’s try a numerical exam-

ple. Let x= 4. If we express the value of e to five decimal places, the left side of the above

equation works out as

(10^4 )(e^4 )≈ 10,000 × 2.71828^4

≈ 10,000 × 54.59800

≈ 545,980

and the right side becomes

(10e)^4 ≈ (10 × 2.71828)^4

≈ 27.1828^4

≈ 545,980

Mixing exponentials in a ratio

How about ratios of mixed common and natural exponentials? If x is a real number, then

10 x / ex= (10/e)x

This is an adaptation of the power of quotient rule from Chap. 9. We can work this out using

x= 4, as in the previous example. Expressing the value of e to five decimal places but rounding

off our final answer to only two decimal places, the left side of the above equation becomes

104 / e^4 ≈ 10,000 / 2.71828^4

≈ 10,000 / 54.59800

≈ 183.16

494 Logarithms and Exponentials