Math Problem Statement

What is the value of x when \(x = 2 \log_5 1 + \log_3 81\)?

Solution

The problem in the image asks for the value of xx when:

x=2log51+log381x = 2 \log_5 1 + \log_3 81

Let's break it down step by step.

Step 1: Simplify the expression log51\log_5 1

The logarithm of 1 in any base is 0, so: log51=0\log_5 1 = 0

Thus, the first term becomes: 2×log51=2×0=02 \times \log_5 1 = 2 \times 0 = 0

Step 2: Simplify log381\log_3 81

We need to express 81 as a power of 3. Since 81=3481 = 3^4, we have: log381=log334=4\log_3 81 = \log_3 3^4 = 4

Step 3: Combine the results

The equation now simplifies to: x=0+4=4x = 0 + 4 = 4

Therefore, the value of xx is 4\boxed{4}.

Would you like more details or have any questions? Here are some related questions you might find interesting:

  1. How do you evaluate logarithms with different bases?
  2. What are the properties of logarithms that are useful in simplifying expressions?
  3. How can you convert logarithms to a different base using the change of base formula?
  4. What is the significance of logarithms in real-world applications?
  5. How would you solve an equation involving a sum of logarithms with different bases?

Tip: Remember that the logarithm of 1 is always 0, regardless of the base, because any number raised to the power of 0 equals 1.

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Math Problem Analysis

Mathematical Concepts

Logarithms
Algebra
Exponents

Formulas

\(\log_b 1 = 0\)
\(\log_b (a^n) = n \cdot \log_b a\)

Theorems

Logarithm properties
Exponentiation properties

Suitable Grade Level

Grades 10-12