Use logarithms to solve the given equation. (Round the answer to four decimal places.)
8x = 11

To solve the equation \( 8^x = 11 \) using logarithms, follow these steps:

### Step 1: Take the logarithm of both sides.
You can take the logarithm of both sides of the equation using any logarithm base, but for convenience, let's use the natural logarithm (\(\ln\)):

\[
\ln(8^x) = \ln(11)
\]

### Step 2: Apply the logarithmic power rule.
The logarithmic power rule states that \(\ln(a^b) = b \cdot \ln(a)\). Applying this to the left side of the equation:

\[
x \cdot \ln(8) = \ln(11)
\]

### Step 3: Solve for \(x\).
Now, solve for \(x\) by dividing both sides by \(\ln(8)\):

\[
x = \frac{\ln(11)}{\ln(8)}
\]

### Step 4: Compute the logarithms and find \(x\).
Using a calculator to find the natural logarithms:

\[
\ln(11) \approx 2.3979
\]
\[
\ln(8) \approx 2.0794
\]

Now, divide the two values:

\[
x \approx \frac{2.3979}{2.0794} \approx 1.1532
\]

### Final Answer:
\[
x \approx 1.1532
\]

Would you like further clarification on any step or more details?

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Here are five related questions for further practice:
1. Solve the equation \( 5^x = 20 \) using logarithms.
2. Solve \( 7^{2x} = 50 \) for \(x\).
3. What is the value of \( x \) if \( 3^x = 15 \)?
4. How would you solve \( 9^{x-1} = 27 \) using logarithms?
5. Can logarithms solve \( e^{2x} = 10 \)? If so, how?

**Tip**: When solving exponential equations with logarithms, remember that any logarithm base (common or natural) will work, but natural logarithms are often preferred for simplicity when using calculators.

Use logarithms to solve the given equation. (Round the answer to four decimal places.) 8^x = 11

Discover how to solve the exponential equation 8^x = 11 using logarithms. This step-by-step solution covers key algebraic concepts such as the logarithmic power rule. Suitable for students in Grades 10-12. Final answer rounded to four decimal places: x ≈ 1.1532.

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