sums will set you free
how to teach your child numbers arithmetic mathematics
writing down logarithms
Logs (or logarithms) to base 10 are normally written as log. Logs to other bases are written as log2 for logs to base 2, or log5 for logs to base 5, and so on. It is important that the number indicating the base of the logarithm is clearly written as smaller and below the word ‘log’, otherwise log2 might be confused for log 2, a very different animal.
In the 1600s, Napier invented logs and Briggs created useable log tables (we will look at these a little later).
From early on, slide rules were developed, and were used widely for science and engineering calculations until the advent of computers and pocket calculators.
Below is a scientific slide rule.
This linear slide rule is set to do a simple calculation - 3 x 2.
The central slider is moved so its number 1 is aligned with the 3 on the upper half of the frame. The left orange diamond in the enlargement below highlights this.
The 2 mark on the central slider lines up with the 6 on the outside frame. So this sum’s answer is 6 (highlighted by the right orange diamond), but why?
Remember that a slide rule is physical version of doing calculations using exponents, or logs. With our sum of 3 x 2, we take the length equivalent to log 3 on the outside frame and add to it the length equivalent to log 2. The resulting combined length is the equivalent to log 6. So although we have added together two lengths, the result is the equivalent of multiplying two numbers.
With the slide set in this position, you can do further calculations such as 3 x 3, or 3 x 2·5 (= 7·5).
As well as linear slide rules, there are (or were, if you believe these tools to be redundant antiques) circular slide rules.
Here is a photo of one, pretty isn’t it?
Instead a frame and a slider, there are inner and outer rings that can move separately. You can just see the transparent cursor block, that can rotate around the slide rule’s centre point. This slide rule has a diameter of 12.3 cm.
As with the linear slide rule we show just above, this slide rule has been set to to show a simple multiplication sum, again 2 x 3 = 6.
On this circular slide rule, the starting points for the different log lengths are marked with a triangle.
With the slide set in this position, you can do further 2-times calculations: 2 x 4 = 8, 2 x 5 = 1 (!). For this last sum, you have to use your intelligence and realise that the zero is not shown, as 2 times 5 equals 10.
Log tables generally come in small books, including as well as log tables, tables for calculating other variables such as cosines, cosecs, cotangents and anti-logs - used to convert a log number back into an integer.,
A bit further down this page are part of four-figure (four-place) log tables. They are used for doing multiplication and division sums using decimal numbers with up to four figures after the decimal point.
Before using the tables, you need to understand some things about them. First be aware that although there is no decimal point in the log tables, the groups of four numbers are decimal fractions.
Next, log tables are designed to save space, so the size of the number is separated from its value. What does all this mean? Well, instead of having separate tables for say 1 to 100, 100 to 1,000 and 1000 to 10,000, there is one set of tables, and the user - you - indicate afterwards the size of the number.
Note that in each case, the characteristic is the same as the index (or exponent) when the number concerned is expressed (written) as a number to a power of ten: 100, 101, 102, 103.
And the something, the part to the right of
Now to do a sum using log tables. Suppose you want to
multiply 24·78 x 33·16. We start by working
out the mantissa, the part to the right of the decimal
point in the log,
This mantissa is added to the characteristic of 24·78 to give the log.
This mantissa is added to the characteristic of 33·16 to give the log.
Now, 24·78 x 33·16 being a multiplication
sum, the two logs are
2.9147 is a log to base ten - that is, it is 102.9147.
Well, this can be done using another set of tables called anti-logarithms, but here we are going to be tough and numerate, and use the log tables themselves!
So the mantissa resolves to 82175.
Checking by calculator, the result of 24·78 x 33·16 = 821·7048. Remember that we have been using four-figure logs, so small inaccuracies will creep in.
So this is the way they used to do it in the olden days, with logs as a short cut (of course, you become a lot faster with practice), or with much faster, but less accurate, slide rules. There are logarithm and other tables going up to ten figures, and probably beyond, which can be useful for checking out computer programmes. And of course, computer programmes can now generate such tables through various iterations.
Now you have your handy pocket calculator, you may think that calculating with logarithms is a bit tiresome. But remember, understanding the processes involved is very useful in building a foundation in understanding how numbers and indices/powers work.
You notice that there all sorts of other scales on the slide rules shown above, allowing you to look up many other interesting ratios and results. Similar look-up tables are generally available in log table books.
When a number between zero and one is converted to a base 10 logarithm, log10, the characteristic of the log (like the power of a number to base 10) will be negative. That is, while a log’s mantissa is always positive, its characteristic will be negative for numbers that are less than 1.0.
When writing by hand, it is easy to write a negative characteristic as it should be, with a line - the bar - above the number. Unfortunately, typical computer keyboards (as well as calculators and modern typesetting) cannot make a mark like this, so a negative characteristic is very often shown as a minus sign before the characteristic. Because we are also limited by computer typesetting capabilities, we also will put a minus sign to indicate a negative characteristic - a bar log. This is what a log with a negative logarithm looks like when hand-written .
When you write down logs, you can either write them
like this example, or you can write them as a calculator
or computer does - if you are sure than you will not become
confused. Remember the characteristic
can be negative or positive, but the mantissa (to the
right of the point or dot) is always positive.
Now this can easily cause confusion because a minus sign in front of a decimal number, such as -1·234 (or nowadays frequently written/typed as -1.234) applies to all the numbers in the decimal: our example decimal number is made up of -1 and - ·234.
A log that is described in words as bar one point two three four, and written as -1.234, is made up of a characteristic of -1 and a mantissa of .234, not -.234.
Here is an example sum: 3 x ·09.
Using the log tables above,
Since 3 x ·09 is a multiplication sum, the two
Disassembling the log into its characteristic and mantissa:
bar 1 and 4313.
Now we use the characteristic of -1 to determine the size of the resolved number. A characteristic of -1 means that the number must be between ·1 and ·9. So the resolved number must be ·2699.
Again rounding errors have crept into our calculation using logs. Doing this sum directly, the answer is ·27.
When multiplying negative numbers, we would carry on as if the numbers were positive and then, at the end of the multiplication, we would correct the final result using the usual rules. That is, positive times positive or negative times negative give a positive result, while negative times positive and positive times negative give a negative result; or, same signs positive, opposite signs negative. This can be expressed in a typical binary table:
For divisions using logs, analogous procedures apply. For a - b (for instance, 3 - 2), the log of 2 will be subtracted from the log of 3. Otherwise, continue as usual. You can also see this process on the slide rule above, where 3 x 2 = 6 can also be read in reverse where 6 ÷ 2 = 3.
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