Indices and Logarithm – SS1 Further Mathematics Lesson Note (First Term)

Indices are powers that show how many times a base is multiplied by itself, and a logarithm is the index to which a base is raised to give a number.

Behavioural objectives

By the end of the lesson, students should be able to:

1. state and apply the laws of indices, including the zero, negative and root indices;
2. solve exponential equations by writing both sides with the same base;
3. convert between index form and logarithmic form;
4. state the laws of logarithm;
5. use the laws of logarithm to simplify and evaluate expressions.

What are the key terms in indices?

A number of the form a^m means a multiplied by itself m times, where a is a real number.

• Base: the number being multiplied, which is a in a^m.
• Index (plural indices): the small raised number m, which tells how many times the base is used.
• Exponent: another name for the index.
• Example: in 2⁵, the base is 2 and the index is 5, so 2⁵ = 2 × 2 × 2 × 2 × 2 = 32.

What are the laws of indices?

There are seven laws of indices: multiplication, division, power, zero index, product power, negative index and root power.

Multiplication law

• Rule: add the indices when the same base is multiplied.
• Long way: p³ × p² = (p × p × p) × (p × p) = p⁵.
• Short way: p³ × p² = p^{3 + 2} = p⁵.

Division law

• Rule: subtract the indices when the same base is divided.
• Example: p⁶ ÷ p⁴ = p^{6 – 4} = p².

Power law

• Rule: multiply the indices when a power is raised to another power.
• Long way: (p³)² = p³ × p³ = p^{3 + 3} = p⁶.
• Short way: p^{3 × 2} = p⁶.

Zero index

• Rule: any non-zero number raised to the power zero is 1.
• Proof, step 1: a^m ÷ a^m = a^{m – m} = a⁰.
• Proof, step 2: a^m ÷ a^m = a^m/a^m = 1.
• Conclusion: so a⁰ = 1.
• Examples: 3⁰ = 1, c⁰ = 1, y⁰ = 1.

Product power law

• Rule: raise each factor in the bracket to the power.
• Example: (2xy)² = 2² × x² × y² = 4x²y².

Negative index

• Rule: a negative index means the reciprocal of the positive power.
• Example 1: 2^-1 = 1/2.
• Example 2: 3^-2 = 1/3² = 1/9.

Root power law

• Rule: an index of 1/n means the nth root of the base.
• Example 1: 9^1/2 = √9 = 3.
• Example 2: 27^1/3 = ³√27 = 3, because 3³ = 27.
• Fractional index: a^m/n = (ⁿ√a)^m, so 8^2/3 = 2² = 4.

How do we solve exponential equations of linear form?

An exponential equation has the unknown in the index, and it is solved by writing both sides with the same base.

• Key idea: if the bases on both sides are equal, then the powers are equal.
• Reverse: if the powers are equal, the bases are equal.

Steps for solving

1. Write each side as a power of the same base (usually a small prime such as 2, 3 or 10).
2. Use the laws of indices to give each side a single power.
3. Equate the indices.
4. Solve the resulting linear equation.

Worked examples

Example 1: (1/2)^x = 8

1. (2^-1)^x = 2³
2. 2^-x = 2³
3. -x = 3
4. x = -3

Example 2: (0.25)^{x + 1} = 16

1. (25/100)^{x + 1} = 4²
2. (1/4)^{x + 1} = 4²
3. (4^-1)^{x + 1} = 4²
4. 4^{-x – 1} = 4²
5. -x – 1 = 2
6. -x = 2 + 1 = 3
7. x = -3

Example 3: 3^x = 1/81

1. 3^x = (1/3)⁴
2. 3^x = 3^-4
3. x = -4

Example 4: 10^x = 1/0.001

1. 10^x = 1000
2. 10^x = 10³
3. x = 3

Example 5: 4/2^x = 64^x

1. 4 ÷ 2^x = 64^x
2. 2² ÷ 2^x = (2⁶)^x
3. 2^{2 – x} = 2^6x
4. 2 – x = 6x
5. 2 = 6x + x = 7x
6. Divide both sides by 7: x = 2/7

What is a logarithm?

The logarithm to base a of a number P is the index x to which a must be raised to equal P.

• Meaning: if P = a^x, then x = log_a P.
• Equivalence: log_a P = x and a^x = P say the same thing.
• Conditions: the base a is positive and not 1, and P must be positive.

What are index form and logarithmic form?

Index form is the way of writing a number as a power, and logarithmic form is the same fact written with log.

How do we convert from index form to logarithmic form?

1. Identify the base, the index and the answer in a^x = P.
2. Write log, with the base as a subscript.
3. Put the answer P after it, then set it equal to the index x.

How do we convert from logarithmic form to index form?

1. Identify the base, the number and the logarithm in log_a P = x.
2. Write the base with the logarithm as its index.
3. Set it equal to the number P.

What are the laws of logarithm?

The laws of logarithm are the product, quotient and power laws, together with log_b b = 1 and log_b 1 = 0.

These laws come from the laws of indices, as the proofs below show.

Product law

1. Let P = b^x, so log_b P = x.
2. Let Q = b^y, so log_b Q = y.
3. PQ = b^x × b^y = b^{x + y} (multiplication law of indices).
4. So log_b PQ = x + y.
5. Therefore log_b PQ = log_b P + log_b Q.

Quotient law

1. P ÷ Q = b^x ÷ b^y = b^{x – y} (division law of indices).
2. So log_b (P/Q) = x – y.
3. Therefore log_b (P/Q) = log_b P – log_b Q.

Power law

1. Pⁿ = (b^x)ⁿ = b^xn (power law of indices).
2. So log_b Pⁿ = nx.
3. Therefore log_b Pⁿ = n log_b P.

Log of the base and log of one

• Log of the base: b = b¹, therefore log_b b = 1.
• Log of one: 1 = b⁰, therefore log_b 1 = 0.

What mistakes should students avoid?

• Adding inside a log: log_b (P + Q) is not equal to log_b P + log_b Q.
• Dividing two logs: log_b P ÷ log_b Q is not log_b (P/Q).
• Different bases: the product, quotient and power laws need the same base throughout.
• Adding bases: (a + b)^m is not equal to a^m + b^m.

How do we use the laws of logarithm to simplify and evaluate?

To simplify a logarithm expression, apply the laws until each term is a log of the base or a given value.

Worked examples

Example 1: log₃ 27 + 2 log₃ 9 – log₃ 54

1. = log₃ 27 + log₃ 9² – log₃ 54
2. = log₃ (27 × 9²/54)
3. = log₃ (27 × 81/54)
4. = log₃ (81/2)
5. = log₃ 3⁴ – log₃ 2
6. = 4 log₃ 3 – log₃ 2
7. = 4 × 1 – log₃ 2
8. = 4 – log₃ 2

Example 2: log₃ 13.5 – log₃ 10.5

1. = log₃ (13.5/10.5)
2. = log₃ (135/105)
3. = log₃ (27/21)
4. = log₃ 27 – log₃ 21
5. = log₃ 3³ – log₃ (3 × 7)
6. = 3 log₃ 3 – (log₃ 3 + log₃ 7)
7. = 3 – 1 – log₃ 7
8. = 2 – log₃ 7

Example 3: log₂ 8 + log₃ 3

1. = log₂ 2³ + log₃ 3
2. = 3 log₂ 2 + log₃ 3
3. = 3 + 1
4. = 4

Example 4: given log₁₀ 2 = 0.3010, log₁₀ 3 = 0.4771 and log₁₀ 5 = 0.699, find log₁₀ 64 + log₁₀ 27

1. = log₁₀ 2⁶ + log₁₀ 3³
2. = 6 log₁₀ 2 + 3 log₁₀ 3
3. = 6(0.3010) + 3(0.4771)
4. = 1.806 + 1.4313
5. = 3.2373

Evaluation

1. State the multiplication, division and power laws of indices.
2. Simplify (a) p⁷ × p² and (b) (p²)⁴ ÷ p³.
3. Solve the equation 3^x = 1/27.
4. Write 5³ = 125 in logarithmic form, and write log₂ 32 = 5 in index form.
5. Use the laws of logarithm to evaluate log₃ 18 – log₃ 2.

Assignment

1. Solve the equation 2^x = 1/16.
2. Write 10³ = 1000 in logarithmic form, and write log₄ 16 = 2 in index form.
3. Given that log₁₀ 2 = 0.3010 and log₁₀ 3 = 0.4771, find log₁₀ 32 + log₁₀ 9.