If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

### Course: Algebra (all content)>Unit 11

Lesson 31: Logarithmic scale (Algebra 2 level)

# Logarithmic scale

Sal discusses the differences between linear and logarithmic scale. Created by Sal Khan.

## Want to join the conversation?

• I'm confused about why Sal just says "multiply by 3 again" to get to 9 (at ""), because if you multiply log(3) by 3 you get 1.43, which is definitely different from log(9)=0.95. If someone could please explain that'd be great--thanks in advance.
• ! is equal to 100 because theres a million numbers in between 1 and 100 same with 1 million
(1 vote)
• in logarithmic scale,why does the length between 0 and 1 is greater than the length between 8 and 9? and also why the length is not uniform throughout?
• If you go from having 9 of something to having 8 of it, you lost 11%, but if you go from having 1 to 0, you lost 100%. Even though you're losing the same number of things, the logarithmic scale reflects that you're losing more on a proportional basis.
• Few points I am wondering about
a) Many a formulas we see log operated with the base 'e' i.e. the natural log. Is there any special significance of this
b) This video explained about the logarithmic scale, it would be very helpful to give more real life examples for use of log (Most of the logarithmic examples in the site just showed how log can be used to simplify calculation eg. taking 6th root of a number)
• At he says "...this is true of ANY logarithmic number line." Does that mean that there is more than one logarithmic scale, so would you call a scale with base x a different scale from one with base y?
Another question: could you have a scale that instead of multiplying or adding by a number x, you take the power x? If you could have that type of scale, what number would you start from? Zero and one won't do anything, so maybe two? Also, then if you were to use Knuth's up arrow notation, could you have different scale for every time you would add an up arrow to the operation? Is this what he means by having more than one logarithmic scale from my first question?
• I will answer the first question. What he meant is that there's various types of logarithmic scales. The most common ones use 10 as the base (like the Richter Scale). But you can find scales like the Krumbein phi scale that uses a log with base two.
• At , did Sal start it at one because you can't use zero in logarithms?
• if he would have had taken zero then the scale was not possible because he multiplies by 10 and again and again,so if there was zero then 0*10=0
hope that helps
• Where is Vi Hart's logarithm video?
• Where would you place a negative number on the logarithmic number line ??
• there cant be a negative no. as root of negative no. is not real but complex so it is not possible!
• Would the scale look the same if instead of base 10 we chose base 5 or base 23?
• I will need to be careful answering this question, since it might be easy to mislead you. I also ask that you read carefully.

First, visualize our scale as being on a ruler, where each number is 1 cm apart.

In a base ten logarithmic scale, 2 cm on the scale corresponds to 10² (100). 3 cm corresponds to 10³(1000). Obviously, if we change the base of the logarithm, the spacing between each number doesn't matter, it is always one cm between each one. However, the same values represented on the scale end up at different distances on the ruler. log base 5 of 100 is about 2.86, and so while on the base 10 scale 100 is 2 cm away from the edge, in base 5 it is 2.86 cm from the edge. Likewise, 1000 was 3 cm away on the base 10 scale, but is now 4.29 cm away.

For your own enrichment, there is a neat property of logarithms that might help you understand what is going on. Note how in the above example, both 100's and 1000's distance from the edge moved by a factor of about 1.43 (meaning its distance was multiplied by 1.43 when we changed scales, 2 cm * 1.43 = 2.86 cm, 3 cm * 1.43 = 4.29 cm). This is due to the following property of logarithms (where log_n means log base n):

log_a(X) = log_b(X) / log_b(a)

This means that if we wish to convert where X is in cm when we change from base b to base a, we multiply by 1/log_b(a), which in our above example is 1/log_10(5) which is 1.43.

If this extra bit confused you, or if I didn't even answer your question to your liking in the first place, go ahead and let me know and I'll get back to you

Cheers