Solve Log28 Without Calculator

What is log28?

The notation log28 represents a logarithm with base 2 of the number 28. In mathematical terms, it asks "To what power must 2 be raised to get 28?"

Since 28 is not a power of 2, log28 is an irrational number approximately equal to 4.807354922057604.

Methods to solve log28 without a calculator

Prime Factorization Method

This method breaks down 28 into its prime factors and then applies logarithm properties.

1. Factorize 28: 28 = 2 × 2 × 7 = 2² × 7¹
2. Apply logarithm properties: log₂(28) = log₂(2² × 7) = log₂(2²) + log₂(7) = 2 + log₂(7)
3. Recognize that log₂(7) is approximately 2.807354922057604
4. Add the values: 2 + 2.807354922057604 ≈ 4.807354922057604

Exponent Estimation Method

This method estimates the exponent by finding powers of 2 near 28.

1. Calculate powers of 2: 2⁴ = 16, 2⁵ = 32
2. Note that 28 is between 16 and 32, so log28 is between 4 and 5
3. Estimate the fractional part by considering 28/16 = 1.75
4. Take the logarithm of 1.75: log₂(1.75) ≈ 0.807354922057604
5. Add to the integer part: 4 + 0.807354922057604 ≈ 4.807354922057604

Step-by-step examples

Example 1: Using Prime Factorization

Calculate log₂(28) using prime factorization:

1. Factorize 28: 28 = 2² × 7
2. Apply logarithm properties: log₂(28) = 2 + log₂(7)
3. Find log₂(7) ≈ 2.807354922057604
4. Final result: 2 + 2.807354922057604 ≈ 4.807354922057604

Example 2: Using Exponent Estimation

Calculate log₂(28) by estimating exponents:

1. Find powers of 2 near 28: 2⁴ = 16, 2⁵ = 32
2. Determine the range: 4 < log₂(28) < 5
3. Calculate 28/16 = 1.75
4. Find log₂(1.75) ≈ 0.807354922057604
5. Combine results: 4 + 0.807354922057604 ≈ 4.807354922057604

Common mistakes to avoid

• Assuming log28 is an integer - it's actually an irrational number
• Incorrectly applying logarithm properties - remember that log(a × b) = log(a) + log(b)
• Using the wrong base - always ensure you're using base 2 for log28
• Rounding too early in calculations - keep more decimal places until the final result