and the right side becomes
(10/e)^4 ≈ (10 / 2.71828)^4≈ 3.67880^4
≈ 183.16
Now let’s invert this ratio. Suppose x is a real number. Then we can write
ex / 10x= (e/10)xAgain, let’s use x= 4 and go through this with a calculator. Expressing the value of e to five
decimal places and leaving our final answer at five decimal places too, the left side of the above
equation works out as
e^4 / 10^4 ≈ 2.71828^4 / 10,000≈ 54.59800 / 10,000
≈ 0.00546
and the right side works out as
(e/10)^4 ≈ (2.71828 / 10)^4≈ 0.271828^4
≈ 0.00546
Are you confused?
To get a “snapshot” of how exponentials of different bases work out, here’s a trick to get the general idea.
First, we determine the values of 1 to the powers of −4,−3,−2,−1, 0, 1, 2, 3, and 4. (That’s right, the base
is 1!) Then we do the same thing with the bases 2, e, and 10. Finally, we compare the values as shown in
Table 29-1. For the bases 1, 2, and 10, all the results are exact. For the base e, we approximate everything
to four decimal places except e^0 , which is exactly 1.
Here’s a challenge!
Find the number whose natural exponential function value is exactly 1,000,000, and the number whose
natural exponential function value is exactly 0.0001.
Solution
To solve this problem, we must be sure that we know what we’re trying to get! Suppose we call the solution
x. In the first case, we can solve the following equation for x:
ex= 1,000,000How Exponentials Work 495