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?

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.