Business Mathematics and Statistics · Class 11 Commerce
Ch 3Logarithm — Class 11 Business Mathematics and Statistics, concept-first.
A logarithm answers a single question: to what power must a fixed base be raised to produce a given number? Before calculators and computers, this idea — introduced by John Napier in the early seventeenth century and refined by Henry Briggs — was the practical tool that turned tedious multiplications, divisions, powers…
Key concepts
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Antilogarithm and Interpolation
The antilogarithm reverses a logarithm: , so if then . It is read from an antilog table using only the mantissa to get the digits, after which the characteristic fixes the decimal point; a negative logarithm must first b…
Most relevant Q&A
Chapter contents
The NCERT structure, section by section. Open a section to see its questions, then read the concept-first solution.
Concept of Logarithm
A logarithm answers a single question: to what power must a fixed base be raised to produce a given number? Before calculators and computers, this idea — introduced by John Napier in the early sevente…
Features and Properties of Logarithm
Several properties follow directly from the definition . They are used constantly in simplification and should be understood, not merely memorised.
Laws of Logarithmic Operations
Three fundamental laws convert the arithmetic of numbers into the arithmetic of their logarithms — turning multiplication into addition, division into subtraction, and powers into multiplication.
Common Logarithm — Characteristic and Mantissa
For numerical work the common logarithm (base 10) is used, and its value is split into two parts:
Determination of Logarithm and Antilogarithm
Finding a logarithm from a log table. A four-figure logarithm table lists the mantissae of digit strings. To find : 1.
Interpolation (the Mean-Difference Method)
A four-figure log table lists mantissae directly only up to the third significant digit. The fourth significant digit is supplied by interpolation — the assumption that, over the tiny interval between…
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 w…
Exercises
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- Q10Find the antilogarithm of $\bar{2}.6596$ (that is, characteristic $-2$, mantissa $0.6596$).Free
- Q13State and prove the three fundamental laws of logarithms (the product, quotient and power laws), taking a common base $a$.Preview
- Q14The value of $\log_{10}1000$ is: (a) 3, (b) 30, (c) 100, (d) $10^{3}$.Preview
More questions
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- Example 1Express the following in the alternative form: (i) write $2^{5}=32$ in logarithmic form, and (ii) write $\log_{3}81 = 4$ in exponential form…Free
- Example 2Evaluate $\log_{2}64$ without using tables.Free
- Example 3Given $\log 2 = 0.3010$ and $\log 3 = 0.4771$ (base 10), find (i) $\log 6$, (ii) $\log 12$, (iii) $\log 5$ and (iv) $\log 1.5$.Free
- Example 4Simplify and evaluate: $2\log 5 + \log 8 - \log 2$ (all to base 10).Preview
- Example 5Evaluate $\log_{8}32$ using the change-of-base rule.Preview
- Example 6Given $\log 2 = 0.3010$, find the number of digits in $2^{50}$.Preview
- Example 7Write down the characteristic of the common logarithm of each of the following numbers: (i) 4567, (ii) 45.67, (iii) 0.4567, (iv) 0.004567.Preview
- Example 8Find $\log 45.67$ using a four-figure logarithm table, showing the mean-difference (interpolation) step for the fourth significant digit.Preview
- Example 9Find the antilogarithm of 2.6596.Preview
- Example 11A sum of ₹5,000 is invested at 8% per annum compound interest for 5 years. Using logarithms (given $\log 1.08 = 0.0334$, $\log 5 = 0.6990$,…Preview
- Example 12Using logarithms, evaluate $\dfrac{25.36 \times 18.5}{6.4}$. (Given $\log 25.36 = 1.4041$, $\log 18.5 = 1.2672$, $\log 6.4 = 0.8062$, and an…Preview