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Logarithms: Class 9 Mathematics

Written by Hassan Raza (Mathematics)

Scientific notation, the definition of a logarithm, common and natural logarithms, characteristic and mantissa, the laws of logarithms, using log tables or a calculator, and applying logarithms to calculations.

Key points

Very large and very small numbers are written in scientific notation as a × 10ⁿ, where 1 ≤ a < 10 and n is an integer. For example 45600 = 4.56 × 10⁴ and 0.00032 = 3.2 × 10⁻⁴. Moving the decimal point to the left gives a positive power, and moving it to the right gives a negative power.

Logarithms were invented to turn long multiplication and division into addition and subtraction. If aˣ = N, where a > 0 and a is not 1, then x is called the logarithm of N to the base a, written x = logₐ N. So the two statements aˣ = N and logₐ N = x are the same fact in two forms.

Examples of the definition: 2⁵ = 32 gives log₂ 32 = 5. 10³ = 1000 gives log₁₀ 1000 = 3. 5⁰ = 1 gives log 1 = 0 for any base. Also logₐ a = 1 for any base a. The logarithm of zero or of a negative number is not defined, because a positive base raised to any power is always positive.

Two bases are used most often. Logarithms to base 10 are called common logarithms, and log N means log₁₀ N. Logarithms to base e (about 2.718) are called natural logarithms and are written ln N. In class 9 we work mainly with common logarithms.

Laws of logarithms (for positive M, N and a base a not equal to 1):

  • Product law: logₐ(MN) = logₐ M + logₐ N.
  • Quotient law: logₐ(M/N) = logₐ M − logₐ N.
  • Power law: logₐ(Mⁿ) = n logₐ M.
  • Change of base: logₐ N = log N / log a, where both logs use the same new base.

These laws follow from the laws of exponents. Let logₐ M = x and logₐ N = y, so M = aˣ and N = aʸ. Then MN = aˣ⁺ʸ, which means logₐ(MN) = x + y. The quotient and power laws are proved in the same way, using aˣ ÷ aʸ = aˣ⁻ʸ and (aˣ)ⁿ = aⁿˣ.

A common mistake is to treat log(M + N) as log M + log N. There is no law for the logarithm of a sum or a difference. Also (log M)(log N) is not log(M + N), and log M ÷ log N is not log(M − N).

The common logarithm of a number has two parts. The integral part is the characteristic and the decimal part is the mantissa, which is always kept positive. For example log 4560 = 3.6590, so the characteristic is 3 and the mantissa is 0.6590.

Rules for the characteristic of log N:

  • If N ≥ 1, the characteristic is one less than the number of digits before the decimal point. So 4560 has characteristic 3, and 45.6 has characteristic 1.
  • If 0 < N < 1, the characteristic is negative and equals minus (one plus the number of zeros right after the decimal point). So 0.0456 has characteristic −2, written 2̄.
  • Numbers with the same digits in the same order have the same mantissa, whatever the position of the decimal point.

The antilogarithm reverses the process. If log x = y, then x = 10ʸ, and we say x is the antilog of y. To find an antilog, use the mantissa to get the digits and use the characteristic to place the decimal point. A calculator uses the 10ˣ key for this and the log key for the logarithm.

To calculate with logarithms: write the expression, take the log of both sides, apply the laws to turn products into sums and quotients into differences, add or subtract the logarithms, and find the antilog of the result. For example, for x = (a × b) / c, we get log x = log a + log b − log c.

Logarithms are also used to solve equations in which the unknown is a power. For 3ˣ = 81 we write 3ˣ = 3⁴, so x = 4. If the bases cannot be matched, take logs of both sides: x log 3 = log 81, so x = log 81 / log 3.

Check the domain before using logarithm laws. For real logarithms, each original argument must be positive and the base must be positive and different from 1. Combining expressions can otherwise hide an invalid candidate solution.

Study tip: Learn the exponential and logarithmic forms as two ways of saying the same thing, and practise switching between them, because most one mark and two mark questions test only this. In long answers, write each law you use by name, show the sum or difference of logs line by line, and always state the characteristic before using tables.

Definitions and flashcards

Say the meaning first, then open the card to check.

Logarithm

The power x to which a base a must be raised to give N, written logₐ N = x.

Base

The number that is raised to a power; it must be positive and not equal to 1.

Common logarithm

A logarithm to base 10.

Natural logarithm

A logarithm to base e, written ln.

Characteristic

The integral part of a common logarithm.

Mantissa

The positive decimal part of a common logarithm.

Antilogarithm

The number whose logarithm is given; if log x = y then x = 10ʸ.

Scientific notation

A number written as a × 10ⁿ with 1 ≤ a < 10.

Formula sheet

NameFormulaWhat it means
Definitionaˣ = N ⇔ logₐ N = xExponential and logarithmic forms are equivalent.
Product lawlogₐ(MN) = logₐ M + logₐ NLog of a product is the sum of logs.
Quotient lawlogₐ(M/N) = logₐ M − logₐ NLog of a quotient is the difference of logs.
Power lawlogₐ(Mⁿ) = n logₐ MBrings the power in front.
Change of baselogₐ N = log N / log aConvert between bases.
Special valueslogₐ 1 = 0 and logₐ a = 1True for every valid base a.

Short questions with model answers

These are the kind of questions asked for 2, 3 and 5 marks. Try each one before opening the answer.

Write 3⁴ = 81 in logarithmic form. (2 marks)

If aˣ = N then logₐ N = x. Here a = 3, x = 4 and N = 81, so log₃ 81 = 4.

What are common logarithms? Which base do they use? (2 marks)

Common logarithms are logarithms to the base 10. The base is usually not written, so log N means log₁₀ N. They are used for calculations because our number system is based on 10.

Why is the logarithm of a negative number not defined? (2 marks)

A logarithm is a power of a positive base. A positive number raised to any real power always gives a positive result, so no power of the base can equal a negative number or zero. Hence logs of negative numbers and zero are not defined.

Prove that logₐ(MN) = logₐ M + logₐ N. (3 marks)

Let logₐ M = x and logₐ N = y. Then M = aˣ and N = aʸ. Multiplying, MN = aˣ × aʸ = aˣ⁺ʸ. By the definition of a logarithm, logₐ(MN) = x + y. Substituting x and y gives logₐ(MN) = logₐ M + logₐ N.

Find the characteristic of log 7380 and of log 0.0073. (3 marks)

7380 has 4 digits before the decimal point, so the characteristic is 4 − 1 = 3. In 0.0073 there are 2 zeros after the decimal point, so the characteristic is −(2 + 1) = −3, written 3̄.

Explain the steps for evaluating x = (23.5 × 4.2) / 7.9 using logarithms. (5 marks)

Take the logarithm of both sides: log x = log 23.5 + log 4.2 − log 7.9. Find each logarithm from tables or a calculator. Add the first two, then subtract the third. This gives log x. Then find the antilog of this value to get x. Check by estimating: 23.5 × 4.2 is about 99 and 99/7.9 is about 12.5, so x should be near 12.5.

What does log_b(a) = c mean? (2 marks)

It means b^c = a, where a and b are positive and b is not 1. A logarithm gives the exponent needed to obtain a.

Explain why log10(100 x 10) equals 3. (3 marks)

The product rule gives log10(100) + log10(10) = 2 + 1 = 3. Equivalently, 100 x 10 = 1000 = 10³.

Solve log2(x - 1) + log2(x + 1) = 3 and check the domain. (5 marks)

Both arguments must be positive, so x > 1. Combine the logs: log2((x - 1)(x + 1)) = 3. Therefore x² - 1 = 8 and x² = 9. The candidates are 3 and -3, but -3 violates the original domain. The only real solution is x = 3.

Numericals with worked steps

Find the value of log₂ 32.
  1. Let log₂ 32 = x, so 2ˣ = 32.
  2. Write 32 as a power of 2: 32 = 2⁵.
  3. So 2ˣ = 2⁵ and x = 5.

Answer: 5

Simplify log 8 + log 125 (base 10).
  1. Use the product law: log 8 + log 125 = log (8 × 125).
  2. 8 × 125 = 1000.
  3. log 1000 = log 10³ = 3 log 10 = 3.

Answer: 3

Solve log_x 64 = 3 for x.
  1. Change to exponential form: x³ = 64.
  2. 64 = 4³.
  3. So x³ = 4³, hence x = 4.

Answer: x = 4

Given log 2 = 0.3010, find log 16 and log 5.
  1. 16 = 2⁴, so log 16 = 4 × log 2 = 4 × 0.3010 = 1.2040.
  2. 5 = 10 ÷ 2, so log 5 = log 10 − log 2 = 1 − 0.3010 = 0.6990.

Answer: log 16 = 1.2040 and log 5 = 0.6990

Solve 3ˣ = 81 and then simplify log₃ 81 − log₃ 9.
  1. 81 = 3⁴, so 3ˣ = 3⁴ and x = 4.
  2. log₃ 81 = 4 and log₃ 9 = 2.
  3. Subtract: 4 − 2 = 2. Check with the quotient law: log₃ (81/9) = log₃ 9 = 2.

Answer: x = 4 and the difference is 2

Solve log10(5x) = 2.
  1. The argument must be positive, so x > 0.
  2. Convert to exponential form: 5x = 10² = 100.
  3. Divide by 5 and check the original argument.

Answer: x = 20; 5x = 100 is positive.

Evaluate log2(8) + log2(4) without a calculator.
  1. Write 8 = 2³ and 4 = 2².
  2. Their logarithms are 3 and 2.
  3. Add 3 + 2.

Answer: 5, also equal to log2(32).

Common mistakes

  • Writing log (M + N) = log M + log N. The product law applies only to a product.
  • Using a base that is zero, negative or equal to 1, or taking the log of zero or a negative number.
  • Making the mantissa negative. Only the characteristic can be negative; the mantissa is always positive.
  • Counting digits wrongly for the characteristic: for numbers above 1 it is digits before the point minus one.
  • Confusing log (Mⁿ) = n log M with (log M)ⁿ, which is a different quantity.

Practise this chapter

Practice questions and timed test Printable worksheet

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