542 Review Questions and Answers
Answer 29-8
The common exponential of a number is what we get when we raise 10 to a power equal to
that number. The natural exponential of a number is what we get when we raise e to a power
equal to that number. When working with natural exponentials, it’s customary to call e the
exponential constant.
Question 29-9
How can we find the number x whose common exponential is 100,000? How can we find the
numbery whose natural exponential is 100,000?
Answer 29-9
To find the number x whose common exponential is 100,000, we must find the power of 10
that gives us 100,000. We want to solve the equation
10 x= 100,000
It’s easy see that x= 5 in this case. But if we want to go through the motions of solving
the above equation formally, we can take the common log (symbolized log 10 ) of both sides,
obtaining
log 10 (10x)= log 10 100,000
The common log function “undoes” the common exponential function, so we can simplify
this equation to
x= log 10 100,000
A calculator tells us that log 10 100,000 is exactly equal to 5.
Finding the number y whose natural exponential is 100,000 is a little more involved. We
want to find the power of e that gives us 100,000, so we must solve the equation
ey= 100,000
If we take the natural log (symbolized ln) of both sides of this equation, we get
ln (ey)= ln 100,000
The natural log function “undoes” the natural exponential function, so we have
y= ln 100,000
A calculator tells us that y= 11.513, rounded off to three decimal places.
Question 29-10
How can we find the number x whose natural exponential is 1/e^5? How can we find the num-
bery whose common exponential is 1/e^5?