Swing the Logarithmic Curve around (1, 0)

The logarithmic function to the base , where and , is defined by if and only if ; the domain is and the range is .
Move the slider; the base of the logarithm changes and you see its graph swing around the point .
Closely observe the two cases and . Also notice where the blue curve lies in relation to the common logarithm (base 10) and the natural logarithm .



  • [Snapshot]
  • [Snapshot]
  • [Snapshot]


When considering the common logarithm (i.e., base 10), we notice that as the values decrease from 1 to 0, the curve falls rapidly, and for , it approaches the negative axis asymptotically. As the values increase from 1 to 10, the function increases monotonically from 0 to 1, and as values increase by a factor of 10 (for example, from 10 to 100) the function increases from 1 to 2. The same applies for the intervals , , and so on. Because the changes are very small for such large intervals, the curve can be well approximated by a straight line.
To switch bases, we let ; we will show that .
By definition, implies .
Taking the to the base of both sides gives .
Dividing by gives . Replacing by yields .
    • Share:

Embed Interactive Demonstration New!

Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details »

Files require Wolfram CDF Player or Mathematica.

Related Curriculum Standards

US Common Core State Standards, Mathematics

Mathematica »
The #1 tool for creating Demonstrations
and anything technical.
Wolfram|Alpha »
Explore anything with the first
computational knowledge engine.
MathWorld »
The web's most extensive
mathematics resource.
Course Assistant Apps »
An app for every course—
right in the palm of your hand.
Wolfram Blog »
Read our views on math,
science, and technology.
Computable Document Format »
The format that makes Demonstrations
(and any information) easy to share and
interact with.
STEM Initiative »
Programs & resources for
educators, schools & students.
Computerbasedmath.org »
Join the initiative for modernizing
math education.
Step-by-Step Solutions »
Walk through homework problems one step at a time, with hints to help along the way.
Wolfram Problem Generator »
Unlimited random practice problems and answers with built-in step-by-step solutions. Practice online or make a printable study sheet.
Wolfram Language »
Knowledge-based programming for everyone.
Powered by Wolfram Mathematica © 2017 Wolfram Demonstrations Project & Contributors  |  Terms of Use  |  Privacy Policy  |  RSS Give us your feedback
Note: To run this Demonstration you need Mathematica 7+ or the free Mathematica Player 7EX
Download or upgrade to Mathematica Player 7EX
I already have Mathematica Player or Mathematica 7+