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Here we show how to construct and use a simple logarithm table to multiply, divide, and take powers of numbers (including fractional powers).

Scientists and engineers need to understand logarithms and exponents as inverse functions, and for that reason, this is a learning resource. This is also an ongoing research project. Genuine research must have both unknown outcome, as well as purpose. The purpose of this research is twofold:

1. In spite almost universal access to calculating software, there will always be a need to quickly perform mental arithmetic. [factual?]
2. The calculator is now an obsolete tool, having been replaced by hand held devices with virtually unlimited access to mathematical knowledge on the internet. These devices pose a problem for those who construct high school and college exams. Instead of forcing students to buy obsolete calculators, why not simply forbid them?

The unknown outcome of this research involves the search for appropriate algorithms appropriate to an era where both slide rules and scientific calculators are obsolete.

## Base-approximately 5/4

What makes this simple logarithmic table unique is that it can be constructed by multiplying a few fractions and making some approximations. By deriving (not memorizing) a simple logarithmic table, you can gain insight into the nature of logarithms and exponents. The following approximation that suggests we take 5/4 as an approximation for the tenth root of 10:

$(5/4)^{10}=9.3132...\approx 10$ To construct our simple log table, take multiple powers of 5/4 and simplify your fractions as you proceed:

${\frac {5}{4}}={\color {Red}1.25\approx 10^{0.1}}\,,$ $\left({\frac {5}{4}}\right)^{2}={\frac {25}{16}}\approx {\frac {24}{16}}={\frac {3}{2}}={\color {Red}1.5\approx 10^{0.2}}\,,$ $\left({\frac {5}{4}}\right)\left({\frac {3}{2}}\right)={\frac {15}{8}}\approx {\frac {16}{8}}={\color {Red}2\approx 10^{0.3}}\,,$ $\left({\frac {5}{4}}\right)\left(2\right)={\frac {5}{2}}={\color {Red}2.5\approx 10^{0.4}}\,,$ $\left({\frac {5}{4}}\right)\left({\frac {5}{2}}\right)={\frac {25}{8}}\approx {\frac {24}{8}}={\color {Red}3\approx 10^{0.5}}\,,$ $\left({\frac {5}{4}}\right)\left(3\right)={\frac {15}{4}}\approx {\frac {16}{4}}={\color {Red}4\approx 10^{0.6}}\,,$ $\left({\frac {5}{4}}\right)\left(4\right)={\color {Red}5\approx 10^{0.7}}\,,$ $\left({\frac {5}{4}}\right)\left(5\right)={\frac {25}{4}}\approx {\frac {24}{4}}={\color {Red}6\approx 10^{0.8}}\,,$ $\left({\frac {5}{4}}\right)\left(6\right)={\frac {30}{4}}\approx {\frac {32}{4}}={\color {Red}8\approx 10^{0.9}}\,,$ $\left({\frac {5}{4}}\right)\left(8\right)={\color {Red}10}\,.$ Three values in the table to the left are inaccurate. Fortunately they can be corrected using a little common sense. Regarding (5/4)2=25/16:

${\frac {24}{16}}={\frac {3}{2}}<{\frac {25}{16}}<{\frac {25}{15}}={\frac {5}{3}}\approx 1.7$ • As a compromise, we replace 1.5→1.6 ≈ 100.2.

The chart to the left has 100.5=3. To get a better estimate of the square root of 10, note that:

$(3)(3.3)=9.9$ • As a compromise, we replace 3→3.2 ≈ 100.5

The chart to the left can be improved by calculating 100.8 starting from 10 (which requires far fewer steps than start

$10^{0.8}={\frac {10}{10^{.2}}}\approx {\frac {10}{(5/4)^{2}}}={\frac {160}{25}}=6.4$ (In the last step, note that dividing by .25 is equivalent to multiplying by 4.)

y=log x 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
x=10y 1.00 1.26 1.58 2.00 2.51 3.16 3.98 5.01 6.31 7.94 10
yx+0.5 1.12 1.41 1.78 2.24 2.82 3.55 4.47 5.62 7.08 8.91 11.2

While adjusting the antilogarithms 1.5→1.6, 3.0→3.2, and 6.0→6.3 slightly enhances precision, it renders the construction (from memory) of the table more problematical.

### Taking powers and roots

Write the number as 10x and modify the exponent as appropriate.

Estimate 601/3:

6  ≈  100.8
60  ≈   101.8
601/3  ≈  100.6  (since 1.8÷3 = .6)
From the chart, 100.6  ≈  4.

601/3  ≈  4 (the exact answer is 3.91...)

### Multiplying and dividing numbers

Add and subtract the exponents as appropriate:

Estimate 2.6  ×  3.7   ÷  7.4

2.6 ≈ 100.4
3.7 ≈ 100.6
7.4 ≈ 100.9
0.4 + 0.6 − 0.9 = 0.1
100.1 ≈ 1.25

2.6 × 3.7 ÷ 7.4 ≈ 1.25 (The exact answer is 1.30)

### Testing the log table as an alternative to multiplication

A preliminary test of this table as an alternative to multiplying and dividing produced unsatisfactory results. Random numbers were used to develop exercises involving multiplication and/or division. One set of problems generated solved problems of the form: ${\frac {1.29}{1.47\cdot 1.41\cdot 2.3}}\approx {\frac {1.25}{1.5\cdot 1.5\cdot 2}}$ $=10^{0.1}\cdot 10^{-0.2}\cdot 10^{-0.2}\cdot 10^{-0.3}$ $=10^{-0.6}=10^{0.4}\cdot 10^{-1}=2.5\cdot 10^{-1}$ This exercise involved four terms, but exercises with two and six terms were also investigated. In all cases it was found that the problems could be more quickly solved using ordinary multiplication, with rounding of terms to one or two significant figures. Only rarely was a division problem encountered that could not quickly be performed the old fashioned way.

### Next topics to study:

Is the log table is the best way to take nontrival powers of numbers? For example:

$17^{3/2}\approx \left(15\right)^{1.5}\approx \left(10^{1.2}\right)^{1.5}\approx 10^{1.8}\approx 70$ Perhaps we need to construct simple multiplication tables that reflect the logarithmic scaling required for accurately multiplying and dividing numbers?

.01.1 1 1.3 1.6 2 2.5 3.2 4 5 6.3 7.9 10
.11.4 1.3 1.6 2 2.5 3.2 4 5 6.3 7.9 10 13
.21.8 1.6 2 2.5 3.2 4 5 6.3 7.9 10 12.6 16
.32.2 2 2.5 3.2 4 5 6.3 7.9 10 13 16 20
.42.8 2.5 3.2 4 5 6.3 7.9 10 13 16 20 25
.53.5 3.2 4 5 6.3 7.9 10 13 16 20 25 32
.64.5 4 5 6.3 7.9 10 13 16 20 25 32 40
.75.6 5 6.3 7.9 10 13 16 20 25 32 40 50
.87.1 6.3 7.9 10 13 16 20 25 32 40 50 63
.98.9 7.9 10 13 16 20 25.1 32 40 50 63 79
111 10 13 16 20 25 32 40 50 63 79 100

The first column indicates the logarithm (0, .1, .2,...) as well as the maximum value that corresponds to a given row. (In other words, 1.05 rounds down to 1.00 and 1.15 rounds up to 1.3)

The reader will recognize that this could be a homemade table since the numbers have already been calculated. Since it is unlikely to find oneself in a position where it is necessary to construct this table by hand, the more accurate numbers are used.

### Sample problem using the logarithmic multiplication table

To divide 64 by 17, round 17 to 16 and 64 to 63. Then search the 16 row for the 64:

${\frac {64}{17}}\approx {\frac {63}{16}}\approx 4$ Note how this could have been easily guessed as approximately 64/16. The idea of this multiplication table is that students could have it as they attempt to work without a calculator. If they ever get stuck they can use the table instead of doing long division.

• Instead of memorizing this multiplication table, it might be better just to practice these tasks until they seem easy.