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To solve the equation:

\[
\log_3(x^5) = -1
\]

we need to convert this logarithmic equation into an exponential form. Here's the step-by-step solution:

### Step 1: Rewrite the Logarithmic Equation
The equation \(\log_3(x^5) = -1\) implies:
\[
x^5 = 3^{-1}
\]

### Step 2: Simplify the Right Side
Since \(3^{-1} = \frac{1}{3}\), we get:
\[
x^5 = \frac{1}{3}
\]

### Step 3: Take the Fifth Root
To solve for \(x\), take the fifth root of both sides:
\[
x = \left(\frac{1}{3}\right)^{\frac{1}{5}}
\]

### Step 4: Simplify Further
This expression can be simplified further, or you can calculate its approximate decimal value:
\[
x \approx 0.8660
\]

So, the solution is:
\[
x \approx 0.8660
\]

Would you like more details or have any questions?

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### Related Questions
1. How would you solve \(\log_4(x^3) = -2\)?
2. What happens if the base of the logarithm is changed to 10 in a similar problem?
3. How would you solve \(\log_5(x^2) = 3\)?
4. What is the general method to solve \(\log_b(x^n) = k\) for \(x\)?
5. Can logarithmic equations have no solution? If so, under what conditions?

### Tip
When solving logarithmic equations, converting them to exponential form is often the most straightforward approach.

Learn how to solve the logarithmic equation log_3(x^5) = -1 with a detailed step-by-step solution. This problem covers logarithmic and exponential properties, showing how to convert between logarithmic and exponential forms. Suitable for high school students in Grades 9-12.

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