AP Notes, Outlines, Study Guides, Vocabulary, Practice Exams and more!

e and the Natural Logarithm

e and the Natural Logarithm

One of the most important numbers used as the base for exponential and logarithmic functions is denoted as e. It is an irrational number, with its value approximated as e » 2.7182818. Exponential and logarithmic functions with base e occur in many practical applications, including those involving growth and decay, continuous compounding of interest, alternating currents and learning curves.

The natural logarithm of a positive real number is defined as the logarithm to the base e of the number. The natural logarithm of x, x > 0, is denoted as ln x. Symbolically,

ln x = loge x where x>0

By definition, ln x = y implies that ey = x.

When converting from base 10 to base e, we can use the following formula:

log x = .4343 ln x

where log 10 e = .4343.

Since the function f(x) = ex and f(x) = ln x are inverse functions of each other,

ln ex = x and eln x = x

The natural logarithm possesses the same properties as common logarithms.

ex + 3 = 17
ln ex + 3 = ln 17
x + 3 = ln 17
x + 3 = 2.833
x = -0.167

ln x + ln (x - 2) = ln 15
ln [x(x - 2)] = ln 15
x2- 2x = 15
x2- 2x - 15 = 0
x2+ 3x -5x - 15 = 0
x(x + 3) -5(x + 3) = 0
(x - 5)(x + 3) = 0

x - 5 = 0 or x + 3 = 0
x = 5 or x = -3

Since x cannot be negative, the solution set is {5}.

Subject X2: 

Need Help?

We hope your visit has been a productive one. If you're having any problems, or would like to give some feedback, we'd love to hear from you.

For general help, questions, and suggestions, try our dedicated support forums.

If you need to contact the Course-Notes.Org web experience team, please use our contact form.

Need Notes?

While we strive to provide the most comprehensive notes for as many high school textbooks as possible, there are certainly going to be some that we miss. Drop us a note and let us know which textbooks you need. Be sure to include which edition of the textbook you are using! If we see enough demand, we'll do whatever we can to get those notes up on the site for you!