# Imprecise logarithms that reference sets of numbers.

I apologize in advance if my question seems vague, I'm only in algebra II, so It may turn out that I lack the terminology to phrase my question correctly.

Some background, we just finished our unit on logarithms, and while I understood the material as presented, I found it to be superficial and unsatisfying. It left me with many questions that i'd like answered.

It seems to me like a logarithm is a number which references a location on a logarithmic scale. When I pick my slide rule, this becomes immediately apparent to me. In my math class, it is nearly always specified that any computed logarithm is merely an approximation, except in cases where the log is a clear, discrete value. As in log 1000 = 3. Usually we round to four significant digits and call it good. However it seems to me that logs lacking in precision actually must reference a set of numbers, and that the higher the level of precision (the more exact the log) the smaller the set becomes.

How many significant digits is considered precise enough?

How can a number referenced too by an irrational log really be said to exist?

Is there a way to understand imprecise logs in terms of tiny little sets of numbers?

How can I come to accept that imprecision is good enough?

• There is nothing special about logarithms in this regard. For example, trigonometric functions have the same problems. To quantify inaccuracies you can use interval arithmetic. See,en.wikipedia.org/wiki/Interval_arithmetic . – jbuddenh Jul 27 '15 at 20:21

The problem has nothing in particular to do with logarithms -- it is a general phenomenon that we cannot always represent the true result of a computation exactly on paper, and so we have to settle for approximations.

For example we have $\log_{10}(13) = 1.1139433523...$. The true, mathematical value of $\log_{10}(13)$ is an exact, precise number which is the number such that 10 to that power is 13 exactly. There's nothing approximate that, except that writing down that number with decimals is not possible; we would need to write an infinity of digits to do that.

But that is not much different from, say, square roots, which I assume you will have seen already. We have $\sqrt{13} = 3.6055512754...$ -- this is again an exact number that we cannot write down exactly, so for practical calculations we have to settle for approximations.

Even plain old arithmetic shows this, for example in division: $1\div 13 = 0.0769230769...$. Here it so happens that the decimals repeat, so we can write exactly $1\div 13 = 0.\overline{076923}$ -- but that is not a very useful representation for further calculations of the number, so in practice we will just choose to cut it off after some number of significant digits.

Taking a ten digit approximation, as I've done in the above examples, gives more than pretty good precision for most everyday purposes. In other words the approximations work quite well for the purposes they're made for: \begin{align} 10^{1.1139433523} &= 12.999999999795350... & (\text{pretty darn close to }13) \\ 3.6055512754^2 &= 12.999999999538566... & (\text{pretty darn close to }13) \\ 0.0769230769\times 13 &= \phantom{1}0.9999999997 & (\text{pretty darn close to }1)\phantom{1} \end{align}

In most cases, as you see here, we get a similar number of correct digits when we undo the operation that produced our approximation as there is in the approximation itself. In the particular case of logarithms, a good rule of thumb is that the number of digits after the decimal point in the approximation of the logarithm is about the same as the total number of correct digit in the antilogarithm.

In higher (abstract) mathematics, the use of approximations is untidy and distracting, just like you have noticed. We usually deal with that by computing as few actual numbers as possible while we're manipulating formulas. Generally, we prefer to give the exact formula for the answer we've produced, such that the reader can decide for himself how many digits he wants to compute it to, if he's interested in the numeric value.

So "the result is $\log_{10}(13)$" is a better exact answer to a question than "the result is $1.1139433523...$", and one is usually expected to leave it in that form unless there's a specific reason to want a decimal representation.

• I can live with that, thanks. – Ajoe Jul 27 '15 at 20:44