solve each exponential equation, 8-lnx^5=0

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?

---

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!

solve each exponential equation, 8 - lnx^5 = 0

Explore how to solve the exponential equation 8 - ln(x^5) = 0 using logarithmic properties and exponential functions. This solution includes steps like applying the power rule for logarithms and rewriting equations in exponential form. Ideal for high school students in Grades 10-12.

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