Use of Logarithms in Commercial Calculations

The practical payoff of everything above is that logarithms replace hard multiplications, divisions, powers and roots of awkward numbers with easy additions, subtractions and multiplications — after which a single antilog reverses the process to give the answer. This is directly useful across commerce and statistics.

The general procedure to evaluate an expression by logarithms is always the same:

1. Take the logarithm of the whole expression, applying the product, quotient and power laws (§3) to break it into a sum and difference of the logarithms of its parts.
2. Read each individual logarithm from the log table (with interpolation, §6).
3. Add and subtract to obtain the logarithm of the final answer.
4. Take the antilogarithm (§5) of that result to recover the answer, placing the decimal point by the characteristic.

Typical commercial applications.

• Products and quotients of large numbers — e.g. evaluating $\dfrac{25.36 \times 18.5}{6.4}$ (Worked Example 12) becomes a matter of $\log 25.36 + \log 18.5 - \log 6.4$ followed by one antilog.
• Compound interest and amount. The compound-amount formula $A = P\left(1+\dfrac{r}{100}\right)^{n}$ contains a power $n$ that is tedious to expand by hand; taking logs turns it into $\log A = \log P + n\log\left(1+\dfrac{r}{100}\right)$, a single multiplication and addition (Worked Example 11).
• Growth, depreciation and geometric means. Population and sales growth rates, written-down-value depreciation, and the geometric mean $\text{GM} = \text{antilog}\left(\dfrac{\Sigma \log x}{n}\right)$ used with growth data (which reappears in the Measures of Central Tendency chapter) are all evaluated by logarithms for exactly the same reason. …