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Before the era of calculators and computers, logarithm tables were used to simplify multiplication, division, raising to powers, and extracting roots. Logarithm tables exploited the properties of logarithms to make these calculations feasible for people without computers or calculators. The basic logarithm properties that logarithm tables utilized are: 1. \(log(a * b) = log(a) + log(b)\) 2. \(log(a / b) = log(a) - log(b)\) 3. \(log(a^n) = n * log(a)\) 4. \(log(\sqrt[n]{a}) = \frac{1}{n} * log(a)\)

Scientific notation is a way to represent significant figures and simplify the expression of very large or very small numbers. A number written in scientific notation is expressed as the product of a number between 1 and 10 (inclusive), and a power of 10. For example, the number \(237600\) can be written as \(2.376 * 10^5\).

Given logarithms for numbers between 1 and 10, one can calculate the logarithm of any number by converting the number to scientific notation. For example, let's calculate the logarithm of \(237600\) with base 10 using logarithm tables: 1. Convert the number to scientific notation: \(237600 = 2.376 * 10^5\) 2. Calculate the logarithm: \(log_{10}(237600) = log_{10}(2.376 * 10^5)\) 3. Apply the logarithm property \(log(a * b) = log(a) + log(b)\): \(log_{10}(2.376 * 10^5) = log_{10}(2.376) + log_{10}(10^5)\) 4. Use the logarithm table to find the logarithm of the number between 1 and 10 (\(2.376\) in this case) and add it to the known value of the logarithm of the power of 10: \(log_{10}(2.376) + 5\) The final result will be the logarithm of the original number. In conclusion, the reason why books of logarithm tables gave logarithms only for numbers between 1 and 10 is that any number can be converted to scientific notation and expressed as a product of a number between 1 and 10 and a power of 10. Using the properties of logarithms, one can then calculate the logarithm of the original number with just the logarithm values of numbers between 1 and 10.