Chapter 10: Logarithms — Part 1: Introduction and Laws of Logarithms

CHAPTER 10: LOGARITHMS

PART 1: INTRODUCTION TO LOGARITHMS AND THE LAWS OF LOGARITHMS

Mathematics HipHop — A-Level Pure Mathematics

Logarithms are closely related to indices. While indices help us express repeated multiplication and powers, logarithms help us determine the power to which a given number must be raised to obtain another number.

For example, since 2³ = 8, we can also say that the logarithm of 8 to base 2 is 3.

In this chapter, we develop a firm understanding of logarithms, their laws and their applications to algebraic problems.

1. Learning objectives

By the end of this lesson, you should be able to:

• Explain the meaning of a logarithm.
• Convert exponential expressions into logarithmic form and vice versa.
• Identify the base and argument of a logarithm.
• State the conditions under which a logarithm is defined.
• Evaluate simple logarithms without a calculator.
• Apply the product, quotient and power laws of logarithms.
• Simplify and expand logarithmic expressions.

2. Meaning of a logarithm

A logarithm is the power or index to which a given base must be raised to produce a particular number.

Consider the following exponential expression:

2³ = 8

This tells us that 2 must be raised to the power 3 to obtain 8.

In logarithmic form, the same statement is written as:

log₂ 8 = 3

We read this as: "The logarithm of 8 to base 2 is 3."

General definition

aˣ = N   ⇔   log_aN = x

Where:

• a is the base.
• N is the number whose logarithm is being found, called the argument.
• x is the logarithm, or the required power.

The base must be positive and different from 1, while the argument must be positive.

3. Converting between exponential and logarithmic forms

The two forms express exactly the same mathematical relationship. The main task is to identify the base, power and result correctly.

Worked Example 1

Express each of the following in logarithmic form:

1. 2⁴ = 16
2. 5³ = 125
3. 10² = 100

Reveal the complete solution

Use the rule:

aˣ = N ⇔ log_aN = x

1. 2⁴ = 16 gives log₂16 = 4.
2. 5³ = 125 gives log₅125 = 3.
3. 10² = 100 gives log₁₀100 = 2.

Worked Example 2

Express each logarithmic statement in exponential form:

1. log₂32 = 5
2. log₃81 = 4
3. log₁₀1000 = 3

Reveal the complete solution

Use:

log_aN = x ⇔ aˣ = N

1. log₂32 = 5 gives 2⁵ = 32.
2. log₃81 = 4 gives 3⁴ = 81.
3. log₁₀1000 = 3 gives 10³ = 1000.

4. Conditions for logarithms

Not every logarithmic expression is defined in the real number system. Three important conditions must be satisfied.

1. The base must be positive: a > 0.
2. The base must not equal 1: a ≠ 1.
3. The argument must be positive: N > 0.

Therefore, log₂8 is defined, but log₂0 and log₂(−8) are not defined in the real number system.

Worked Example 3

Determine whether each expression is defined in the real number system:

1. log₃27
2. log₁8
3. log₅(−25)
4. log₄0
5. log₋₂8

Reveal the complete solution

1. log₃27: Defined. The base is positive and different from 1, and the argument is positive.
2. log₁8: Not defined. The base cannot equal 1.
3. log₅(−25): Not defined in the real number system because the argument is negative.
4. log₄0: Not defined because the argument must be positive.
5. log₋₂8: Not defined in the real logarithmic system because the base must be positive.

5. Evaluating simple logarithms

To evaluate a logarithm, ask yourself: "To what power must the base be raised to produce the argument?"

Worked Example 4

Evaluate:

1. log₂64
2. log₃1
3. log₅125
4. log₁₀0.01
5. log₄(1/16)

Reveal the complete solution

1. Since 2⁶ = 64, log₂64 = 6.
2. Since 3⁰ = 1, log₃1 = 0.
3. Since 5³ = 125, log₅125 = 3.
4. Since 10⁻² = 0.01, log₁₀0.01 = −2.
5. Since 4⁻² = 1/16, log₄(1/16) = −2.

6. Important logarithmic identities

Some logarithmic values can be determined directly from the definition.

Identity 1: Logarithm of 1

log_a1 = 0

This is because a⁰ = 1.

Identity 2: Logarithm of the base

log_aa = 1

This is because a¹ = a.

Identity 3: Logarithm of a power of the base

log_a(aˣ) = x

Identity 4: Raising the base to a logarithm

a^logₐN = N

Worked Example 5

Evaluate:

log₇1 + log₇7 + log₇(7⁴)

Reveal the complete solution

Using the identities:

log₇1 = 0

log₇7 = 1

log₇(7⁴) = 4

Therefore:

0 + 1 + 4 = 5

Answer: 5

7. The product law of logarithms

The logarithm of a product is equal to the sum of the logarithms of the individual factors, provided they are expressed using the same base.

log_a(MN) = log_aM + log_aN

Why does the product law work?

Suppose:

log_aM = p   and   log_aN = q

Then:

M = aᵖ   and   N = aᑫ

Multiplying:

MN = aᵖ × aᑫ = a^p+q

Taking logarithms to base a:

log_a(MN) = p + q

Hence the product law follows.

Worked Example 6

Evaluate:

log₂8 + log₂4

Reveal the complete solution

Using the product law:

= log₂(8 × 4)

= log₂32

Since 2⁵ = 32:

= 5

Answer: 5

8. The quotient law of logarithms

The logarithm of a quotient is equal to the logarithm of the numerator minus the logarithm of the denominator.

log_a(M/N) = log_aM − log_aN

Worked Example 7

Evaluate:

log₃81 − log₃9

Reveal the complete solution

Using the quotient law:

= log₃(81/9)

= log₃9

Since 3² = 9:

= 2

Answer: 2

9. The power law of logarithms

The logarithm of a number raised to a power is equal to the power multiplied by the logarithm of the number.

log_a(Mⁿ) = n log_aM

Worked Example 8

Simplify:

log₂(8³)

Reveal the complete solution

Apply the power law:

= 3log₂8

Since 2³ = 8:

= 3 × 3

= 9

Answer: 9

10. Expanding logarithmic expressions

The product, quotient and power laws can be used together to expand complicated logarithmic expressions.

Worked Example 9

Expand fully:

log_a(x²y³/z)

Assume all variables and the base satisfy the conditions for logarithms.

Reveal the complete solution

Step 1: Apply the quotient law.

= log_a(x²y³) − log_az

Step 2: Apply the product law.

= log_ax² + log_ay³ − log_az

Step 3: Apply the power law.

= 2log_ax + 3log_ay − log_az

Answer:

2log_ax + 3log_ay − log_az

11. Condensing logarithmic expressions

Condensing is the reverse of expanding. We combine separate logarithmic terms into a single logarithm by using the logarithmic laws.

Worked Example 10

Express as a single logarithm:

2log₃x + log₃y − log₃z

Reveal the complete solution

Step 1: Apply the power law in reverse.

= log₃x² + log₃y − log₃z

Step 2: Combine the first two terms using the product law.

= log₃(x²y) − log₃z

Step 3: Apply the quotient law.

= log₃(x²y/z)

Answer: log₃(x²y/z)

12. Common mistakes to avoid

1. Confusing the base and argument: In log₂8, the base is 2 and the argument is 8.
2. Forgetting the base: Logarithms with different bases cannot be combined directly using the product or quotient laws.
3. Adding instead of subtracting: The quotient law requires subtraction.
4. Applying the power law incorrectly: The exponent becomes a coefficient in front of the logarithm.
5. Taking the logarithm of zero or a negative number: These are not defined in the real number system.
6. Assuming log(M + N) = log M + log N: This is false. The product law applies to multiplication, not addition.
7. Forgetting brackets: Clearly indicate the complete argument of a logarithm.

13. Student exercises

Attempt all questions before opening the answers. Show your working clearly.

Exercise A: Basic understanding

1. Express 3⁴ = 81 in logarithmic form. [1 mark]
2. Express log₅125 = 3 in exponential form. [1 mark]
3. State the base and argument of log₇49. [2 marks]
4. Evaluate log₂128. [2 marks]
5. Evaluate log₃(1/27). [2 marks]
6. Evaluate log₁₀0.001. [2 marks]

Exercise B: Conditions and identities

1. State the three conditions required for log_aN to be defined in the real number system. [3 marks]
2. Evaluate log₉1 + log₉9 + log₉81. [3 marks]
3. Explain why log₂(−4) is not defined in the real number system. [2 marks]
4. Evaluate log₅(5⁷). [2 marks]

Exercise C: Laws of logarithms

1. Evaluate log₂8 + log₂16. [2 marks]
2. Evaluate log₃81 − log₃3. [2 marks]
3. Simplify log₅(25³). [2 marks]
4. Expand log_a(x³y²). [2 marks]
5. Expand log_m(p⁴/q²). [3 marks]
6. Express 3log₂x + 2log₂y − log₂z as a single logarithm. [3 marks]

Exercise D: Examination-style problems

1. Given that log₂x = 4, find x. [2 marks]
2. Given that log₃y = −2, find y. [2 marks]
3. Express log₅a + 2log₅b − 3log₅c as a single logarithm. [3 marks]
4. Given that log₂x = 3 and log₂y = 5, evaluate log₂(xy²). [3 marks]
5. Expand fully log_k(a²b³/c⁴). [3 marks]

14. Answers to exercises

Exercise A — Answers

1. log₃81 = 4
2. 5³ = 125
3. Base = 7; argument = 49.
4. 7
5. −3
6. −3

Exercise B — Answers

1. a > 0, a ≠ 1, N > 0.
2. 0 + 1 + 2 = 3.
3. The argument is negative, and real logarithms require a positive argument.
4. 7

Exercise C — Answers

1. 3 + 4 = 7
2. 4 − 1 = 3
3. 6
4. 3log_ax + 2log_ay
5. 4log_mp − 2log_mq
6. log₂(x³y²/z)

Exercise D — Answers

1. x = 16
2. y = 1/9
3. log₅(ab²/c³)
4. 3 + 2(5) = 13
5. 2log_ka + 3log_kb − 4log_kc

15. Part 1 revision checklist

Before proceeding to Part 2, make sure you can confidently:

• Define a logarithm using an exponential expression.
• Convert between exponential and logarithmic forms.
• Identify the base and argument.
• State the conditions for a logarithm to exist.
• Evaluate logarithms using powers.
• Apply the product, quotient and power laws.
• Expand logarithmic expressions.
• Condense several logarithmic terms into one expression.

Final message to the learner

Logarithms are not a completely new type of mathematics. They are another way of expressing powers. Once you understand the relationship between indices and logarithms, their laws become much easier to remember and apply.

Practise the examples, understand each law and always pay attention to the base and the argument.

MATHEMATICS HIPHOP — MAKING MATHEMATICS EASIER, ONE CONCEPT AT A TIME.