HP 33S Advanced uses of logarithmic functions


ME33S Advanced uses of logarithmic functions

Log and antilog functions

Before calculators like the ME33S became easily available, logarithms were commonly used to simply multiplication. They are still used in many subjects, to represent large numbers, as the results of integration, and even in number theory.

The ME33S has four functions for calculations with logarithms.

These are the “common” logarithm of “x”, $, its inverse, !, the “natural” logarithm of “x”, & and its inverse, #.
Common logarithms are also called “log to base 10” and the common logarithm of a number “x” is written
LOG10 x or just LOG x
Natural logarithms are also called “log to base e” and the natural logarithm of a number “x” is written
LOGe x or LN x

Logarithms can be calculated to other bases, for example the log to base two of x is written
LOG2 x

Some problems need the logarithm of a number to a base n, other than 10 or e. On the ME33S these can be calculated using one of the formulae

LOGn x = LOG10 x ÷ LOG10 n
LNn x = LNe x ÷ LNe n
! and # are also called “antilogarithms” or “antilogs”. # is also called the “exponential” function or “exp”. Apart from being the inverses of the log functions, they have their own uses. ! is very useful for entering powers of 10, especially in programs where the } key can not be used to enter a power that has been calculated. # is used in calculations where exponential growth is involved. 1# is a quick way to type the value of e.
The ) function can be seen as the base “n” antilog function. If 10x is the inverse of log10 x and ex is the inverse of loge x, then yx is the inverse of logy x.

Practice using log and antilog functions

Example 1: Find the common logarithm of 2.
Solution: In RPN mode type 2¹$

Figure 1

In algebraic mode type 2¹$Ï MEcalculators - 2 - ME33S Advanced uses of logarithmic functions - Version 1.0
ME33S Advanced uses of logarithmic functions

Figure 2

Answer: The common logarithm of 2 is very nearly 0.3010.

Example 2: A rare species of tree has a trunk whose cross-section changes as 1/x with the height x. (Obviously this breaks down at ground level and at the tree top.) The cross section for any such tree is given by A/x, where A is the cross-section calculated at 1 meter above the ground. What is the volume of the trunk between 1 meter and 2 meters above ground?
Solution: The volume is obtained by integrating the cross-section along the length, so it is given by the integral:

Figure 3

It is possible to evaluate this integral using the ME33S integration function, but it is much quicker to note that the indefinite integral of 1/x is LN x. The result is therefore
V = A (LN2 – LN1)
Since LN 1 is 0, this simplifies to
In RPN or algebraic mode type 2&. In algebraic mode follow this with Ï
No one is likely to measure tree heights to an accuracy of more than three significant digits, so set the
ME33S to display the answer with just 3 digits after the decimal point, by pressing Þ13

Figure 4

Answer: Figure 4 shows that the log to base e of 2 is close to 0.693, so the volume is 0.693A cubic meters.

Example 3: What is the log to base 3 of 5? Confirm the result using the ) function.
Solution: Using the equations given above, the log to base 3 of 5 can be calculated as (log10 5)/(log10 3).
In RPN mode, press: 5¹$3¹$¯
In algebraic mode, press: 5¹$¯3¹$Ï
MEcalculators - 3 - ME33S Advanced uses of logarithmic functions - Version 1.0
*12* 33S Advanced uses of logarithmic functions

Figure 5

That this is correct can be confirmed if the following keys are pressed.

In RPN mode: 3w)
In algebraic mode: 3)¹ÍÏ

Figure 6

Answer: The log to base 3 of 5 is 1.465 within the current accuracy setting of the calculator, as shown by Figure 5. Calculating 3 to this power gives 5.000 which confirms that the correct value for the log had been obtained.
Example 4: An activity of 200 is measured for a standard of Cr51 (with a half-life of 667.20 hours). How much time
will have passed when the activity measured in the sample is 170? The formula for half-life computations
is shown in Figure 7.

Figure 7

Solution: Rearrange the equation to solve for t, as in Figure 8.

Figure 8
Now calculate t. In RPN mode:
In algebraic mode:

Figure 9

Answer: 156.4352 hours. Figure 9 shows the result in RPN mode.
Note: When you have finished this example, press Þ4 to return the display to Standard display mode.

Note: Please ignore the above information, in case it has been informed or circulated.

Edited: 10 Nov 2009, 2:53 p.m. after one or more responses were posted


Looking at the not really matching characterset, I presume that this article has been copied from the Net or some text book. Could you point us to the source of your information, please?


Hi, Nasser;

It seems to me that part of the information you posted was copied and pasted from this document, available at the HP Calculators homepage. I thought you would like to know about this, maybe you are not aware of it. The actual HP33S page has many other papers about other subjects. Hope you find them useful as well.


Luiz (Brazil)

Edited: 2 Nov 2009, 4:33 a.m.


Hi Luiz,

Thank you for pointing out.

I knew that this topic is already posted on HP homepage and I have copied it. Perhaps, as mentioned in the NOTE.

I gust wanted to give a little start in this forum.

Reminder : Please ignore it.


Edited: 2 Nov 2009, 11:44 p.m.


... I think all others would like to know more about you. You see, your 'first appearance' was quite unusual, and I believe most 'old timers' agree with me. As a bare comparison, if this is an automotive-address museum you were introducing yourself by showing us the best way to change tires... by using data from a well known document with no regards mentioning the source. Can you understand why did you toke us by surprise?

Now I´d invite you to tell us about which calculators you use, program, etc... I began contributing here in 2001, and people are totally friendly. But as Namir mentioned, we are not babes in the woods. At least not the most of us... 8^)


Luiz (Brazil)

Edited: 3 Nov 2009, 3:57 a.m.


Naseer, you wrote above:

I knew that this topic is already posted on HP homepage and I have copied it.
But then you answered Michael Meyers with:
Please ignore it, if publicly available.
I do not want to go ahead discussing what seems to be already pointed out, but still, would you like to comment on that?

Luiz (Brazil)

Edited: 3 Nov 2009, 8:25 a.m.


Hi, Nasser;

I see that the rest of the information you posted seems to be copied and pasted from this other document, available at the same HP Calculators homepage I pointed previously.


Luiz (Brazil)

Edited: 2 Nov 2009, 4:52 a.m.


This entire thread adds no value whatsoever and should be deleted. What's the point in pasting info readily available from the HP website?
Jeff Kearns


I second your opinion Jeff. This thread is mega-useless!!!!



There are people on this forum who more and more often comment on posts that they are useless or that the subject has already been discussed and should not be brought up again. I think such comments should be kept away from here since they add very little and may scare people of from the forum, stopping them from posting again. This does not mean that constructive critisism isn't sometimes in place, but that's something else.

There are often subjects here that do not interest me, but I have a simple solution to that - I don't read them. I hope this forum is not a place for a few elite HP-users, but rather a place where also beginners are welcome.


Perhaps one of these (link)



Edited: 2 Nov 2009, 2:35 p.m.


Hi Jeff,

Its true...

Reminder: Please ignore it.



We are not babes in the wood in need for trivial information like the one you posted.



Final reply....

Reminder Note: Please ignore it.


Could be some trolling, but maybe Naseer was just trying to dive in.*

If you're trying to meet people here and make friends, just tell us a bit about yourself and your interests. Do you collect or program any other HP calculators?

"Naseer Norris is a registered user.

Date Registered: November 2009
Location: INDIA
Interests: I like reading science fiction books and I love making friends"


Hi Michael,

Yes, I am a registered user on this forum and trust me, there is nothing like people called trolling.

Thank god, some is here to understand me. I really appreciate you, as i have recently registered with this forum and I wanted to give a start, by posting some information on this forum.

Hope everything is clear.

Reminder: Please ignore it, if publicly available.

Edited: 2 Nov 2009, 11:54 p.m.


Hi Naseer,

The real point is that you could've done something like posting, say, "I found this on the hp website (link) and thought it rather curious, specifically (citation etc)."

By simply posting a copy, with no text of *your own*, it looks like clutter. It is clutter. There is no point to it.

Please see it from other people's perspective--that no, most people do not understand you.

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