what is the value of log subscript 3 baseline 81

Breiz Heurope

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11-03-2025

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The expression log₃ 81 asks: "To what power must we raise the base 3 to obtain 81?" This is a fundamental question in understanding logarithms. Let's explore how to solve it and understand the underlying concept.

Understanding Logarithms

A logarithm is essentially the inverse operation of exponentiation. The expression logₐ b means "the exponent to which we must raise the base a to get b." In other words:

If aˣ = b, then logₐ b = x

In our case, we have log₃ 81. This translates to: "To what power (x) must we raise 3 to get 81?"

Solving log₃ 81

We can solve this problem in a couple of ways:

Method 1: Prime Factorization

Let's break down 81 into its prime factors:

81 = 3 x 3 x 3 x 3 = 3⁴

Therefore, we can rewrite 81 as 3⁴. Substituting this back into our original logarithm:

log₃ 81 = log₃ (3⁴)

Since the logarithm and the exponent are inverse operations, they cancel each other out, leaving us with:

log₃ (3⁴) = 4

Method 2: Using the Change of Base Formula

The change of base formula allows us to change the base of a logarithm to a more convenient base, such as base 10 or base e (natural logarithm). The formula is:

logₐ b = (logₓ b) / (logₓ a)

where x is any valid base. Let's use base 10:

log₃ 81 = (log₁₀ 81) / (log₁₀ 3)

Using a calculator:

log₁₀ 81 ≈ 1.908 log₁₀ 3 ≈ 0.477

log₃ 81 ≈ 1.908 / 0.477 ≈ 4

While this method is useful for logarithms with bases that aren't easily factored, the prime factorization method is often quicker and more intuitive for simpler problems like this one.

Answer and Conclusion

The value of log₃ 81 is $\boxed{4}$. This means that 3 raised to the power of 4 equals 81 (3⁴ = 81). Understanding logarithms and their relationship to exponents is crucial for many areas of mathematics and science. Remember to practice using both methods to solidify your understanding.