We are tasked with solving the equation:

\[
3 \log_x(64) = 6
\]

### Step-by-step solution:

1. **Isolate the logarithmic term:**

   \[
   \log_x(64) = \frac{6}{3} = 2
   \]

2. **Convert the logarithmic equation to exponential form:**

   The logarithmic expression \(\log_x(64) = 2\) implies:

   \[
   x^2 = 64
   \]

3. **Solve for \(x\):**

   Taking the square root of both sides:

   \[
   x = \pm \sqrt{64} = \pm 8
   \]

However, since \(x\) represents the base of a logarithm, it must be positive. Therefore, we discard the negative value.

Thus, the solution is:

\[
x = 8
\]

Would you like further details or have any questions?

Here are 5 related questions you could explore:
1. What is the general method to solve logarithmic equations like \(\log_b(y) = c\)?
2. How does the change of base formula for logarithms work?
3. What if the logarithmic base \(x\) is not restricted to positive values?
4. Can logarithms with negative bases ever be valid?
5. What is the derivative of a logarithmic function in terms of its base?

**Tip:** Always remember that the base of a logarithm must be positive and cannot be 1.

Learn how to solve the logarithmic equation 3log_x(64)=6 with a detailed step-by-step solution. This problem covers key mathematical concepts, such as logarithms, exponentiation, and algebra. Suitable for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation