Math Problem Statement

Solve

Solution

To solve log93\log_9 3, we want to determine the exponent xx such that:

9x=39^x = 3

We can express 9 as 323^2. Therefore, the equation becomes:

(32)x=3(3^2)^x = 3

Simplifying, we have:

32x=313^{2x} = 3^1

Since the bases are the same, we can equate the exponents:

2x=12x = 1

Solving for xx:

x=12x = \frac{1}{2}

Thus, log93=12\log_9 3 = \frac{1}{2}. The correct answer is 12\frac{1}{2}.

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Related Questions:

  1. How do you solve logarithms with different bases?
  2. What is the change of base formula in logarithms?
  3. How do you evaluate log42\log_4 2?
  4. What is the relationship between logarithms and exponents?
  5. How do you simplify expressions involving logarithms?
  6. What is the value of log981\log_9 81?
  7. How do you solve logarithmic equations with multiple logarithms?
  8. What are common logarithms and how are they used?

Tip: Remember that logarithms can be converted to a different base using the change of base formula: logba=logcalogcb\log_b a = \frac{\log_c a}{\log_c b}.

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Math Problem Analysis

Mathematical Concepts

Logarithms
Exponents
Change of Base Formula

Formulas

\( \log_b a = \frac{\log_c a}{\log_c b} \)

Theorems

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Suitable Grade Level

Grades 10-12