SS1 Further Mathematics First Term: Logarithms

Contents

1. Laws of Logarithms
2. Change of Base of Logarithms
3. Use of Tables (Greater than One and Less than One)
4. Logarithmic Equations

Sub Topic: Laws of Logarithms

Introduction to Logarithms

In the previous topic on indices, we learned that if p = a^x (for example, 1000 = 10^3), where 3 is the exponent or index, we can express this relationship in logarithmic form. Thus, the logarithm of p to the base a is defined as follows:

log_a(p) = x or log_10(1000) = 3

Definition of Logarithms

The logarithm of a number p to base a (where a is a positive number not equal to 1) is the exponent to which a must be raised to yield p. This shows that indices and logarithms are closely connected.

Examples:

• log_2(8) = 3 because 8 = 2^3
• log_3(9) = 2 because 9 = 3^2

Laws of Logarithms

1. Multiplication Rule: log_a(pq) = log_a(p) + log_a(q) Example: If log_3(6 * 5) = log_3(6) + log_3(5)
2. Division Rule: log_a(x/y) = log_a(x) – log_a(y)
3. Power Rule: log_a(x^n) = n * log_a(x)
4. Logarithm of 1: log_a(1) = 0
5. Logarithm of the Base: log_a(a) = 1
6. Change of Base Formula: log_b(a) = log_k(a) / log_k(b) for any base k.

Example Problems

1. Simplify:
log_3(9) + log_3(21) – log_3(7)

Solution: log_3(9) + log_3(21) – log_3(7) = log_3(9 * 21 / 7) = log_3(27) = log_3(3^3) = 3

2. Solve for x:
4 * log_x(5) = log_5(x)

Solution: Rearranging gives us: 4 * log_x(5) = 4 / log_5(x) => 4 = log_5(x) * log_x(5)

Taking square roots, we find: x = 25 or x = 1/25.

Class Activities

1. Simplify the following:
   • (a) log_3(27) + 2 * log_3(9)
   • (b) log_x(x^9)
   • (c) log(5)
2. Solve the following logarithmic equations:
   • (i) log_10(x^2 + 4) = 2 + log_10(x) – log_10(20)
   • (ii) log_2(2n) – 2 * log_8(n) = 4

Sub Topic: Change of Base of Logarithms

How to Change the Base of Logarithms

To change the base of logarithms, follow these steps:

Let log_a(x) = y, then x = a^y.

We can change the base as follows: log_b(x) = log_a(x) / log_a(b)

Example:

Convert the following logarithms to base 10:

1. log_3(81) = log_10(81) / log_10(3) = 4 * log_10(3) / log_10(3) = 4
2. log_5(125) = log_10(125) / log_10(5) = 3 * log_10(5) / log_10(5) = 3
3. log_c(x) = d => d = log_10(x) / log_10(c)

Class Activity

• Show that: log_2(8) = 3 by changing the base.

Sub Topic: Use of Tables (Greater than One and Less than One)

Understanding Logarithm Tables

Logarithm tables are essential tools for calculating logarithms, especially for values that are not straightforward.

• Characteristic: The integer part of the logarithm.
• Mantissa: The decimal part of the logarithm.

For example, consider log_10(530), log_10(53), and log_10(5.3):

• log_10(530) = 2.7243 (Characteristic: 2, Mantissa: 0.7243)
• log_10(53) = 1.7243
• log_10(5.3) = 0.7243

Example Using Tables

To evaluate 65.43 * 145.3:

1. Convert to standard form:
   • 65.43 = 6.543 * 10^1
   • 145.3 = 1.453 * 10^3
2. Look up logarithms:
   • log_10(65.43) ≈ 1.815
   • log_10(145.3) ≈ 3.1623
3. Add the logarithms: log_10(65.43 * 145.3) = 1.815 + 3.1623 = 4.9773
4. Convert back to standard form: Antilog(0.9773) gives approximately 9.507 Therefore, 65.43 * 145.3 ≈ 95070.

Class Activities

• Use tables to solve the following:
  • (a) 23.82 * 142.8
  • (b) 0.03167 * 102.8 * 0.325
  • (c) 14.87 / 2.314
  • (d) (12.31)^2
  • (e) 33.28 / 4.689

Sub Topic: Logarithmic Equations

Example 1: Solve for m

Solve the equation: -4(2^(m + 3)) = 4(2^(m + 3)) = 5m + 1

Solution: Equating the powers: -12 – 1 = 5m + 8m -13 = -13m => m = 1

Example 2: Solve for x and y

Given:

1. 8 = y^x – x
2. 4 = y^x + x

Solution: From these equations, we can equate and solve for x and y simultaneously.

Class Activities

• Find n if:
• Given logarithmic equations, solve for x and y.

Practice Exercise

Objective Test

1. Evaluate log_0.258
   • (a) 1/2
   • (b) 2/3
   • (c) -2/3
   • (d) -3/2
2. If 2 * log_4(2) = x + 1, find x:
   • (a) -2
   • (b) -1
   • (c) 0
   • (d) 1
3. Given that log_3(x – y) = 1 and log_3(y) = 0, find x.

Conclusion

In this first term, we’ve covered the fundamentals of logarithms, their laws, methods to change the base, and how to utilize logarithm tables for calculations. The subsequent topics will build upon these foundational concepts to develop a deeper understanding of logarithmic equations and their applications.

Evaluation Questions

1. The logarithm of a number p to base a is defined as: a) log_a(p) = p^a
   b) log_a(p) = x
   c) log_a(p) = a^p
   d) log_a(p) = a^x
2. Which of the following is a law of logarithms? a) log_a(p + q) = log_a(p) + log_a(q)
   b) log_a(pq) = log_a(p) – log_a(q)
   c) log_a(p/q) = log_a(p) + log_a(q)
   d) log_a(pq) = log_a(p) + log_a(q)
3. The characteristic of a logarithm is: a) The decimal part of the logarithm
   b) The integer part of the logarithm
   c) The base of the logarithm
   d) The exponent in the logarithm
4. If log_2(16) = x, then x is: a) 2
   b) 3
   c) 4
   d) 5
5. What is the value of log_5(125)? a) 2
   b) 3
   c) 4
   d) 5
6. The equation log_a(1) = x implies: a) x = 0
   b) x = 1
   c) x = a
   d) x = -1
7. Which of the following represents the power rule of logarithms? a) log_a(m + n) = log_a(m) + log_a(n)
   b) log_a(m/n) = log_a(m) – log_a(n)
   c) log_a(m^n) = n * log_a(m)
   d) log_a(m) = 1/log_b(m)
8. If log_10(100) = x, then x is: a) 1
   b) 2
   c) 3
   d) 4
9. Which of the following is the correct formula for the change of base? a) log_b(x) = log_a(x) + log_a(b)
   b) log_b(x) = log_a(x) – log_a(b)
   c) log_b(x) = log_a(x) / log_a(b)
   d) log_b(x) = log_a(b) / log_a(x)
10. If 2 * log_2(8) = x, then x is: a) 3
    b) 6
    c) 8
    d) 4
11. Which of the following is true for logarithmic equations? a) log_a(x + y) = log_a(x) + log_a(y)
    b) log_a(x – y) = log_a(x) – log_a(y)
    c) log_a(xy) = log_a(x) + log_a(y)
    d) log_a(x/y) = log_a(x) + log_a(y)
12. The logarithm of 1 to any base is: a) 1
    b) 0
    c) -1
    d) Undefined
13. The mantissa of a logarithm is: a) The integer part of the logarithm
    b) The decimal part of the logarithm
    c) The exponent in the logarithm
    d) The logarithm itself
14. If log_3(x) = 2, then x is: a) 6
    b) 8
    c) 9
    d) 3
15. Which of the following logarithmic values is less than 1? a) log_2(4)
    b) log_3(3)
    c) log_5(0.5)
    d) log_10(10)

Class Activity Discussion

1. What is a logarithm?
   • A logarithm is the exponent to which a base must be raised to produce a given number.
2. How is the logarithm of a product expressed?
   • The logarithm of a product can be expressed as the sum of the logarithms: log_a(pq) = log_a(p) + log_a(q).
3. What is the relationship between logarithms and indices?
   • Logarithms are the inverse operations of exponentiation, where log_a(p) = x means that a^x = p.
4. What is the power rule in logarithms?
   • The power rule states that log_a(m^n) = n * log_a(m).
5. How do you change the base of a logarithm?
   • To change the base, use the formula: log_b(x) = log_a(x) / log_a(b).
6. What does the characteristic of a logarithm represent?
   • The characteristic is the integer part of a logarithm.
7. What is the value of log_a(a)?
   • The value of log_a(a) is 1, as any number raised to the power of 1 is itself.
8. How is the logarithm of 1 defined?
   • The logarithm of 1 to any base is always 0 because any number raised to the power of 0 equals 1.
9. What is a mantissa in logarithms?
   • The mantissa is the decimal part of the logarithm, which, along with the characteristic, gives the complete value of the logarithm.
10. How do logarithmic equations differ from simple equations?
    • Logarithmic equations involve logarithmic expressions, which require properties of logarithms to solve, often transforming them into exponential form.
11. What is the value of log_10(10^3)?
    • The value is 3 because 10 raised to the power of 3 equals 10^3.
12. Can logarithms be negative?
    • Yes, logarithms can be negative when the argument (the number for which you are finding the log) is between 0 and 1.
13. Why are logarithm tables useful?
    • Logarithm tables are useful for simplifying calculations, especially before the advent of calculators, by providing quick access to logarithmic values.
14. What happens to the logarithm if the base increases?
    • If the base increases while the argument remains constant, the logarithmic value decreases.
15. How do you evaluate log_2(32)?
    • log_2(32) equals 5 because 2^5 = 32.

Evaluation

1. Evaluate log_2(64):
   Answer: 6
2. If log_5(x) = 2, what is x?
   Answer: 25
3. What is log_10(0.1)?
   Answer: -1
4. Solve for x in the equation 3 * log_10(x) = 1:
   Answer: 10^(1/3)
5. Evaluate log_4(16):
   Answer: 2
6. If log_2(8) = x, what is x?
   Answer: 3
7. What is log_10(1000)?
   Answer: 3
8. Find the value of log_7(49):
   Answer: 2
9. If log_x(27) = 3, find x:
   Answer: 3
10. Evaluate log_3(81):
    Answer: 4