Algebra Know-It-ALL

Part Three 543

Answer 29-10

To find the number x whose natural exponential is 1/e^5 , we must find the power of e that gives
us 1/e^5. We want to solve the equation

ex= 1/e^5

This is almost trivial, because 1/e^5 is just another way of writing e−^5. Now we have

ex=e−^5

Obviously, this means x=−5. If, despite the simplicity of this, we insist on solving formally
and including every step, we can take the natural log of both sides of the above equation,
obtaining

ln (ex)= ln (e−^5 )

We can simplify both sides to get

x=− 5

Finding the number y whose common exponential is 1/e^5 requires more work, but not much.
We want to find the power of 10 that gives us e−^5 , so we must solve the equation

10 y=e−^5

If we take the common log of both sides of this equation, we obtain

log 10 (10y)= log 10 (e−^5 )

The common log function “undoes” the common exponential function, so we have

y= log 10 (e−^5 )

A calculator tells us that e−^5 = 0.006737947, rounded off to nine decimal places. That ought
to be plenty of digits to give us a good idea of the final answer, which is the common log of
0.006737947. Rounding off the end result to three decimal places, we get

y= log 10 (0.006737947)

=−2.171