To solve the equation:

$8 - \ln(x^5) = 0$

we can follow these steps:

Step 1: Isolate the logarithmic term

Move $\ln(x^5)$ to the right side of the equation:

$\ln(x^5) = 8$

Step 2: Use the logarithmic power rule

The power rule of logarithms states that $\ln(a^b) = b \ln(a)$. Applying this rule, we get:

$5 \ln(x) = 8$

Step 3: Solve for $\ln(x)$

Divide both sides by 5:

$\ln(x) = \frac{8}{5}$

Step 4: Exponentiate both sides

To solve for $x$, rewrite the equation in exponential form:

$x = e^{\frac{8}{5}}$

Final Answer

So, the solution is:

$x = e^{\frac{8}{5}}$

This is the exact answer. For an approximate decimal value, you can calculate:

$x \approx 4.953$

Would you like more details or have any other questions?

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1. How does the power rule for logarithms work in different contexts?
2. How do we solve exponential equations involving other bases?
3. How can we convert logarithmic equations to exponential form?
4. What steps are involved when solving for a variable in a natural logarithm?
5. How does this solution method compare to other methods of solving exponential equations?

Tip: Remember, rewriting logarithmic expressions using rules like the power rule can simplify solving complex logarithmic equations!