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what is the value of log subscript 3 baseline 81

what is the value of log subscript 3 baseline 81

2 min read 05-02-2025
what is the value of log subscript 3 baseline 81

What is the Value of log₃ 81? A Step-by-Step Explanation

Title Tag: Value of log₃ 81: A Simple Explanation

Meta Description: Learn how to solve logarithmic equations! This guide breaks down how to find the value of log₃ 81, explaining the concept of logarithms and providing a step-by-step solution. Perfect for students and anyone looking to improve their math skills.

Logarithms might seem intimidating at first, but they're really just a different way of expressing exponents. Understanding this core concept makes solving problems like finding the value of log₃ 81 straightforward. In this case, we're asking: "To what power do we raise 3 to get 81?"

Understanding Logarithms

The expression log₃ 81 is a logarithm. It can be rewritten in exponential form as:

3x = 81

Where 'x' represents the value we're trying to find. In simpler terms, this equation asks: "What number (x) do we need to raise the base (3) to, to get the result (81)?"

Solving for x: A Step-by-Step Approach

  1. Prime Factorization: Let's break down 81 into its prime factors. 81 is divisible by 3, repeatedly:

    • 81 ÷ 3 = 27
    • 27 ÷ 3 = 9
    • 9 ÷ 3 = 3
    • 3 ÷ 3 = 1

    Therefore, 81 = 3 x 3 x 3 x 3 = 34

  2. Substitution: Now, substitute this value back into our original equation:

    3x = 34

  3. Equating Exponents: Since the bases are the same (both are 3), we can equate the exponents:

    x = 4

Therefore, the value of log₃ 81 is 4.

Checking Our Answer

To verify our solution, we can plug the value of 'x' back into the exponential form:

34 = 81

This confirms our answer.

Logarithms in Different Bases

It's important to note that the base of the logarithm (the small number written below the 'log') is crucial. A different base would result in a different answer. For example, log₁₀ 81 would have a different value than log₃ 81.

Practical Applications of Logarithms

Logarithms are used extensively in various fields, including:

  • Science: Measuring earthquake magnitudes (Richter scale), decibel levels (sound intensity), and pH levels (acidity/alkalinity).
  • Finance: Calculating compound interest and determining the growth of investments.
  • Computer Science: Analyzing algorithms and solving computational problems.

This article provides a basic understanding of logarithms and demonstrates how to calculate the value of log₃ 81. Understanding the relationship between logarithms and exponents is key to mastering this mathematical concept. Remember to practice with different examples to build confidence and proficiency.

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