# Error in approximation to log(223)/log(3) .... senior moment?

• B
• Swamp Thing
In summary, the conversation discusses a calculation involving logarithms and an error that arises when using different approaches. The error is shown to be smaller when the calculation is done with larger numbers. The speaker admits to possibly making a mistake and not being able to find it.
Swamp Thing
This is probably a silly question, but I am really stuck. A possible senior moment, is my only excuse.

Here is an approximation:
##log(223)/log(3) \approx 10818288 / 2198026 ##

So we have:
##log(223)/log(3) - 10818288 / 2198026 = 0.0399292##
which is OK but not great -- the error shows up right at the second decimal.

But when we do this:
##10818288 \times log(223) - 2198026 \times log(3)## it gives us -0.000984652, which looks way better.

I would expect the error between two large numbers to be larger than when the same thing is recast as a difference between two small numbers. Again, it's probably a silly thing that I'm missing, but I haven't been able to find it.

Oops, the 0.0399 is not correct, it is actually around 4E-10. I was printing out 5 or 6 things and picked the wrong value from the output list.

## 1. What is the meaning of "error in approximation" when calculating log(223)/log(3)?

The error in approximation refers to the difference between the exact value and the approximate value of log(223)/log(3). It is a measure of how accurate the approximation is compared to the true value.

## 2. Why is there a "senior moment" mentioned in the question?

The term "senior moment" is often used colloquially to refer to a lapse in memory or a mistake made by someone who is older. In this context, it is likely referring to the fact that the calculation of log(223)/log(3) may be more challenging for someone who is not as familiar with logarithms or has not used them in a while.

## 3. How is the error in approximation calculated for log(223)/log(3)?

The error in approximation can be calculated by finding the difference between the approximate value and the exact value of log(223)/log(3), and then dividing by the exact value. This gives a percentage or decimal representation of the error.

## 4. Can the error in approximation for log(223)/log(3) be reduced?

Yes, the error in approximation can be reduced by using more accurate methods of calculation or by increasing the number of decimal places used in the approximation. However, it is important to note that even with a small error, the approximation may still be considered accurate for practical purposes.

## 5. What factors can contribute to a larger error in approximation for log(223)/log(3)?

Factors that can contribute to a larger error in approximation include using a less accurate method of calculation, using a smaller number of decimal places in the approximation, or making a mistake in the calculation process. Additionally, if the exact value of log(223)/log(3) is a repeating decimal, the error may be larger due to rounding.

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