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To solve the logarithmic equation

\[
\log_{729} x = \frac{2}{3}
\]

we need to rewrite it in exponential form to solve for \( x \).

### Step 1: Rewrite in Exponential Form
The equation \(\log_{729} x = \frac{2}{3}\) implies:
\[
x = 729^{\frac{2}{3}}
\]

### Step 2: Simplify the Expression
Now, let's simplify \( 729^{\frac{2}{3}} \).

1. **Express 729 as a power:** We know that \( 729 = 3^6 \).
   So,
   \[
   729^{\frac{2}{3}} = (3^6)^{\frac{2}{3}}
   \]

2. **Use the Power Rule \((a^m)^n = a^{m \cdot n}\):**
   \[
   (3^6)^{\frac{2}{3}} = 3^{6 \cdot \frac{2}{3}} = 3^4
   \]

3. **Calculate \(3^4\):**
   \[
   3^4 = 81
   \]

### Solution
Thus, the positive solution for \( x \) is:
\[
x = 81
\]

Would you like more details on any specific step, or do you have any questions?

### Related Questions:
1. How do you convert a logarithmic equation to its exponential form?
2. What is the power rule in exponents, and how does it apply here?
3. Can we solve this equation using a calculator for verification?
4. How do we determine the base of the logarithm if it isn’t explicitly given?
5. What are some properties of logarithms that can simplify similar problems?

### Tip:
When solving logarithmic equations, rewriting in exponential form is often the most direct way to isolate the variable.

Solve the following logarithm problem for the positive solution for x: log_{729} x = 2/3

Learn how to solve the logarithmic equation log_{729} x = 2/3 by converting it into exponential form and simplifying the expression. This problem covers key logarithmic and exponential concepts suitable for high school students.

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