Basic Logarithmic Equations

Overview

• Focus on solving basic logarithmic equations.

Examples of Solving Logarithmic Equations

1. Log Base 2 of 16 = x

• Convert to exponential form: 2^x = 16
• Solution:
  • 2^4 = 16 → x = 4
  • Alternate method: log(16)/log(2) = 4

2. Log Base x of 81 = 4

• Exponential form: x^4 = 81
• Finding value of x:
  • Fourth root of 81 = 3 → x = 3

3. Log Base 5 of x = 3

• Exponential form: 5^3 = x
• Calculation:
  • 5^3 = 125 → x = 125

4. Log Base 32 of x = 4/5

• Exponential form: 32^(4/5) = x
• Calculation Steps:
  • Fifth root of 32 = 2
  • 2^4 = 16 → x = 16

5. Log Base 3 of (5x + 1) = 4

• Exponential form: 3^4 = 5x + 1
• Calculation Steps:
  • 81 = 5x + 1
  • 5x = 80 → x = 16

6. Log x = 24

• Assume base 10: 10^24 = x
  • x = 10^24

7. Natural Log (ln) x = 7

• Base e: e^7 = x
  • Approximate value: x ≈ 1096.65

8. Log Base 7 of (x^2 + 3x + 9) = 2

• Exponential form: 7^2 = x^2 + 3x + 9
• Calculation Steps:
  • 49 = x^2 + 3x + 9
  • Rearrange: x^2 + 3x - 40 = 0
  • Factor: (x + 8)(x - 5) = 0 → x = -8, 5

9. Natural Log (3x - 2) = 5

• Convert to exponential form: e^5 = 3x - 2
• Rearranging: 3x = e^5 + 2 → x = (e^5 + 2)/3
  • Approximate value: x ≈ 50.14

10. Solve 4 ln(2x - 1) + 3 = 11

• Steps:
  • 4 ln(2x - 1) = 8 → ln(2x - 1) = 2
  • Exponential form: 2x - 1 = e^2 → x = (e^2 + 1)/2
  • Approximate value: x ≈ 4.1945

Solving Logarithmic Equations with Same Base

Example: Log Base 3 (5x + 2) = Log Base 3 (7x - 8)

• Equate inside: 5x + 2 = 7x - 8
• Rearranging: 2x = 10 → x = 5

Quadratic Equations from Logarithmic Equations

Example: Log Base 2 of (x^2 + 4x) = Log Base 2 of 5

• Equate: x^2 + 4x = 5 → x^2 + 4x - 5 = 0
• Factor: (x + 5)(x - 1) = 0 → x = -5, 1

Example: Log Base 2 of x + Log Base 2 of (x + 4) = 5

• Combine: Log Base 2 of (x(x + 4)) = 5
• Exponential form: 2^5 = x^2 + 4x → 32 = x^2 + 4x
• Rearrange: x^2 + 4x - 32 = 0 → (x + 8)(x - 4) = 0 → x = -8, 4
  • Check for extraneous solutions: -8 is invalid for logs, valid solution is x = 4.

Additional Logarithmic Operations

Example: Log Base 3 of (x + 1) = 3 - Log Base 3 of (x + 7)

• Move log: Log Base 3 of (x + 1) + Log Base 3 of (x + 7) = 3
• Combine: Log Base 3 of ((x + 1)(x + 7)) = 3
• Exponential form: 3^3 = (x + 1)(x + 7)
• Solve quadratic: 0 = x^2 + 8x - 27 → x = -10, 2

Example: Log Base 2 (x + 3) - Log Base 2 (x - 3) = 4

• Combine: Log Base 2 of ((x + 3)/(x - 3)) = 4
• Exponential form: 2^4 = (x + 3)/(x - 3)
• Solve:
  • Cross-multiply and rearrange to get x = 5.

Advanced Logarithmic Equations

Example: Log of (log x) = 4

• Exponential form: 10^4 = log x → log x = 10000.
• Second exponential: 10^10000 = x.

Example: Log Base 3 (log Base 2 x) = 2

• Exponential form: 3^2 = log Base 2 x → log Base 2 x = 9.
• Second exponential: x = 2^9 = 512.

Conclusion

• Practice solving these types of equations to gain proficiency in logarithmic functions and exponential transformations.