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Solve for Log 243 Without Calculator

Reviewed by Calculator Editorial Team

Calculating logarithms without a calculator requires understanding the mathematical properties of logarithms and applying them to the given number. This guide explains how to solve for log 243 using different methods.

How to solve log 243 without a calculator

There are several methods to find the logarithm of 243 without a calculator. The most common approaches are:

  • Prime factorization method
  • Exponent method
  • Using common logarithm (base 10)
  • Using natural logarithm (base e)

Each method has its advantages depending on the context and the base of the logarithm you're working with.

Prime factorization method

The prime factorization method involves breaking down the number into its prime factors and then expressing it as a power of a base number.

Formula: If a number can be expressed as \( n = b^k \), then \( \log_b n = k \).

Step-by-step solution for log 243

  1. Factorize 243 into its prime factors:
    • 243 ÷ 3 = 81
    • 81 ÷ 3 = 27
    • 27 ÷ 3 = 9
    • 9 ÷ 3 = 3
    • 3 ÷ 3 = 1
  2. Count the number of times you divided by 3: 5 times
  3. Therefore, 243 = 3^5
  4. So, log₃ 243 = 5

Note: This method works best when the number can be easily expressed as a power of a small integer.

Exponent method

The exponent method involves finding a number that, when raised to a power, equals the original number.

Formula: Find \( k \) such that \( b^k = n \).

Example for log 243

  1. Start with 3^1 = 3
  2. Multiply by 3: 3^2 = 9
  3. Multiply by 3: 3^3 = 27
  4. Multiply by 3: 3^4 = 81
  5. Multiply by 3: 3^5 = 243

Since 3^5 = 243, log₃ 243 = 5.

Common logarithm (base 10)

When working with common logarithms (base 10), you can use the change of base formula.

Change of base formula: \( \log_{10} 243 = \frac{\log_3 243}{\log_3 10} \)

Since we already know log₃ 243 = 5, we can calculate log₃ 10 ≈ 2.0959 (using a calculator for this step).

Therefore, log₁₀ 243 ≈ 5 / 2.0959 ≈ 2.386.

Natural logarithm (base e)

For natural logarithms (base e), you can use the same change of base formula.

Change of base formula: \( \ln 243 = \frac{\log_3 243}{\log_3 e} \)

We know log₃ 243 = 5 and log₃ e ≈ 1.1036 (using a calculator for this step).

Therefore, ln 243 ≈ 5 / 1.1036 ≈ 4.532.

FAQ

What is the value of log 243?
The value of log 243 depends on the base. For base 3, log₃ 243 = 5. For base 10, log₁₀ 243 ≈ 2.386. For natural logarithm (base e), ln 243 ≈ 4.532.
Can I solve log 243 without a calculator?
Yes, you can solve log 243 without a calculator using prime factorization, exponent method, or change of base formula. The prime factorization method is the most straightforward for this specific number.
What is the difference between log, ln, and lg?
log typically refers to base 10 (common logarithm), ln refers to natural logarithm (base e), and lg can sometimes refer to base 2. Always check the base when working with logarithms.
Why is log 243 equal to 5?
Because 243 can be expressed as 3^5, and the logarithm is the exponent to which the base must be raised to get the number. So, log₃ 243 = 5.