I need to create a logarithmic scale that will link an object's "bulk
value" to its volume in liters.

What is "bulk value"? Tentatively I have defined two bits of data to
serve as a guide:

For a 0.04-liter golf ball, bulk value = 1
For a 75-liter human being, bulk value = 20

The following graph should clarify the kind of logarithmic scale I
would like to create.

http://www.gregboettcher.com/graphics/bulkvalue.gif

I need an equation that will let me figure out the volume if I know
the bulk value. And I need an equation to figure out the bulk value if
I know the volume.

I also need you to explain how you got these equations, so I can
modify them on my own (for example, if I want to make it so that a
human being has a bulk value of 10 instead of 20, while a golf ball's
bulk value remains the same).

Finally, I'll give you a tip if you can point me to any resource
(hopefully on the web) that tells me the volume of a wide variety of
common objects (basketballs, beach balls, tin cans, pencils, apples,
etc.). No problem if you can't find any such resource, but I'll gladly
tip you if you can.

Greg

Hi gregwb-ga,

The basic formula for a logarithmic scale would be:

   BV = ln(VOL)

where

   BV is the bulk value number
   VOL is the volume in liters
   ln is the natural logarithm (also known as "log-to-the-base-e")

However, to make this curve fit your selected bulk values (for the
golf ball and human being) we need to include a multiplying factor "a"
and an adding factor "b", like this:

   BV = a ln(VOL) + b

To find out which numbers to use for 'a' and 'b', we need to "plug in"
your two specific values, to get a pair of simultaneous equations:

   1 = a ln(0.04) + b
   20 = a ln(75) + b

We now solve these simultaneous equations for 'a' and 'b', which
yields the following result:

   a = (20 - 1) / ( ln(75) - ln(0.04) )
   b = 1 - (a ln(0.04))

Calculating these values gives:

   a = 2.52111
   b = 9.11514

So your final formula is

   BV = 2.52111 ln(VOL) + 9.11514

You can easily check this formula for your golf ball and human. You
will find that it gives Bulk Values of 0.999999 and 20.000002
respectively, which is as near to 1 and 20 as we can get with this
level of precision.

Now we can use that formula to see, for example, that the bulk value
of an eight-liter beach ball is given by:

   BV = 2.52111 ln(8) + 9.11514 = 14.35764

By the way, a logarithmic scale doesn't "bottom out" at zero. The bulk
value would be negative for very small volumes. That reflects the
usefulness of a logarithmic scale, because the volumes from, say, 0.01
to 0.1 liters are considered to be just as interesting as the volumes
from, say, 10 to 100 liters.

Now, what if you want to adjust your scale to use a different set of
"standard objects" instead of the golf ball and the human? Remember
these two formulas from above...

   a = (20 - 1) / ( ln(75) - ln(0.04) )
   b = 1 - (a ln(0.04))

The numbers in those formulas (0.04, 1, 75, 20) come from the volume
and bulk value of your golf ball and human. For other standard
objects, just change the corresponding values.

For example, if you want an 8-liter beach ball to have BV 3 and a
2000-liter car to have BV 17 just calculate 'a' and 'b' in the
following formulas:

   a = (17 - 3) / ( ln(2000) - ln(8) )
   b = 3 - (a ln(8))

This gives

   a = 2.53556
   b = -2.27255

which we can plug into our BV formula...

   BV = a ln(VOL) + b

...to get...

   BV = 2.53556 ln(VOL) - 2.27255

Any values you calculate with that version of the formula will be
consistent with your beach ball (BV 3) and car (BV 17).

Unfortunately, I don't have any web resource that lists the volume of
common objects. That would indeed be an interesting page to see.

If any of the above doesn't make sense, please request clarification.

Regards,
eiffel-ga

Thanks for your help. The answer was quite different from what I expected!

Thanks, gregwb-ga, for your comments and kind tip.

Regards,
eiffel-ga