Algebra Know-It-ALL

Suppose you have a calculator that can’t directly raise one number to the power of another,

but it does have a common log key. You can calculate eπ using the rule for converting a power

to a product:

log 10 eπ=π log 10 e

Take the value 2.71828 as an approximation of e, and 3.14159 as an approximation of π.

Then, going to five decimal places in each step, the above equation becomes

log 10 eπ≈ 3.14159 log 10 2.71828

≈ 3.14159 × 0.43429

≈ 1.36436

When you take the common antilog of this, you’ll get the value of eπ, because

antilog 10 (log 10 eπ)=eπ

Calculating, you should get

antilog 10 1.36436 ≈ 23.13982

This makes intuitive sense. Think of it like this: Because π is a little more than 3, and because e

is a little less than 3, the value of eπ should be somewhere near 3^3 , which is 27. It’s not terribly

close, but it’s “in the ball park”!

If your calculator has a key for raising one number to the power of another (in general),

you can check this out. When I input the numbers into my calculator, I get

2.718283.14159≈ 23.14058

As usual, rounding error has crept into this process. If you like, you can repeat this exercise,

letting your calculator keep all the extra digits it can along the way, and rounding off to five

decimal places when you get to the very end.

Here’s a challenge!

Using a calculator, find πe. Use the same process as you did to find eπ in the example you just finished.

Round off the values to five decimal places. Verify this result by using the “x^y” key if your calculator

has one.

Solution

Once again, you can take advantage of the rule for converting a power to a product. This time, you have

log 10 πe=e log 10 π

490 Logarithms and Exponentials