How to Solve 100 Log 2 Without Calculator

Understanding Logarithms

A logarithm is the inverse operation of exponentiation. The expression "log_b a" asks, "To what power must the base b be raised to obtain a?" For example, log₂ 8 = 3 because 2³ = 8.

In the expression 100 log 2, we're calculating the logarithm of 100 with base 2. This means we're finding the exponent to which 2 must be raised to get 100.

Calculating 100 log 2

To calculate 100 log 2 without a calculator, we need to find the exponent x such that 2^x = 100. This requires understanding the powers of 2 and using logarithmic identities.

The exact value of log₂ 100 is irrational, but we can approximate it using known powers of 2 and the change of base formula.

Step-by-Step Method

1. Identify the powers of 2 that bracket 100:
   • 2⁶ = 64
   • 2⁷ = 128

   Since 64 < 100 < 128, we know 6 < log₂ 100 < 7.

2. Use linear approximation between known points:

   Linear Approximation Formula

   log₂ 100 ≈ 6 + (100 - 64) / (128 - 64)

   log₂ 100 ≈ 6 + 36 / 64 ≈ 6.5625

3. Refine the approximation using logarithms of known values:

   Refined Approximation

   log₂ 100 ≈ 6.64385619

Verification

To verify our approximation, we can calculate 2^6.64385619 and check if it's close to 100.

Using the exponential function: 2^6.64385619 ≈ 100.000000

This confirms our approximation is accurate to several decimal places.

Common Mistakes

• Confusing log₂ 100 with ln 100 or log₁₀ 100
• Using incorrect powers of 2 for bracketing
• Applying linear approximation without sufficient refinement
• Rounding too early in the calculation process