Solve the equation 2^(3x−1) = 36 for x.

To solve the equation \( 2^{(3x - 1)} = 36 \), follow these steps:

### Step 1: Take the natural logarithm of both sides
Applying the natural logarithm to both sides of the equation helps bring the exponent down:
\[
\ln(2^{(3x - 1)}) = \ln(36)
\]
Using the logarithmic property \( \ln(a^b) = b \ln(a) \), this becomes:
\[
(3x - 1) \ln(2) = \ln(36)
\]

### Step 2: Solve for \( 3x - 1 \)
Now, isolate \( 3x - 1 \):
\[
3x - 1 = \frac{\ln(36)}{\ln(2)}
\]
We can calculate \( \frac{\ln(36)}{\ln(2)} \):
\[
\ln(36) \approx 3.5835 \quad \text{and} \quad \ln(2) \approx 0.6931
\]
So,
\[
\frac{\ln(36)}{\ln(2)} = \frac{3.5835}{0.6931} \approx 5.17
\]
Thus, we have:
\[
3x - 1 = 5.17
\]

### Step 3: Solve for \( x \)
Now, solve for \( x \):
\[
3x = 5.17 + 1 = 6.17
\]
\[
x = \frac{6.17}{3} \approx 2.06
\]

### Final Answer:
\[
x \approx 2.06
\]

Would you like further details or have any questions?

Here are 5 related questions:
1. How do you solve logarithmic equations in general?
2. What are the properties of logarithms?
3. How can logarithms help in solving exponential equations?
4. What if the base of the exponent were a different number?
5. How can you approximate logarithms without a calculator?

**Tip**: Remember that taking the logarithm of both sides is a useful tool for solving exponential equations where the variable is in the exponent.

Learn how to solve the exponential equation 2^(3x - 1) = 36 by applying logarithmic properties. Follow this step-by-step guide to understand the use of natural logarithms in isolating the exponent and solving for x. Suitable for students in Grades 10-12.

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