Algebra Know-It-ALL

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?