solve for x 6^3x + 1 = 2^2x-3 leave your answer in exact form and do it like you would grade 12 math

To solve the equation \( 6^{3x + 1} = 2^{2x - 3} \) for \( x \), we'll follow the steps typically taught in Grade 12 math.

### Step 1: Take the logarithm of both sides
To solve for \( x \), take the natural logarithm (ln) of both sides:

\[
\ln(6^{3x + 1}) = \ln(2^{2x - 3})
\]

### Step 2: Apply the logarithm power rule
Use the power rule of logarithms, which states that \( \ln(a^b) = b \cdot \ln(a) \), to move the exponents in front:

\[
(3x + 1) \cdot \ln(6) = (2x - 3) \cdot \ln(2)
\]

### Step 3: Expand both sides
Distribute the logarithms:

\[
3x \cdot \ln(6) + \ln(6) = 2x \cdot \ln(2) - 3 \cdot \ln(2)
\]

### Step 4: Collect like terms
Group the terms involving \( x \) on one side and the constant terms on the other:

\[
3x \cdot \ln(6) - 2x \cdot \ln(2) = -\ln(6) - 3 \cdot \ln(2)
\]

### Step 5: Factor out \( x \)
Factor \( x \) out from the left-hand side:

\[
x \cdot (3 \ln(6) - 2 \ln(2)) = -\ln(6) - 3 \ln(2)
\]

### Step 6: Solve for \( x \)
Finally, divide both sides by \( (3 \ln(6) - 2 \ln(2)) \) to isolate \( x \):

\[
x = \frac{-\ln(6) - 3 \ln(2)}{3 \ln(6) - 2 \ln(2)}
\]

This is the exact form of the solution for \( x \).

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Would you like any more details, or do you have any questions about the steps? 

Here are 8 related questions you might consider:
1. How do you evaluate the expression \( \ln(6) \) without a calculator?
2. Can you simplify the expression \( 3 \ln(6) - 2 \ln(2) \) further?
3. How would you solve the equation using a different base for the logarithms?
4. What is the significance of using the natural logarithm (ln) in this context?
5. How can you check if your solution for \( x \) is correct?
6. What would happen if the bases of the exponents were the same?
7. How would you express the final solution in decimal form?
8. How would you solve a similar equation where the exponents are quadratic?

**Tip:** Always ensure your final answer is in its simplest exact form before converting to a decimal, if necessary.

Learn how to solve the equation 6^(3x + 1) = 2^(2x - 3) for x with a detailed step-by-step solution. This problem involves exponential functions, logarithms, and equations, suitable for Grade 12 mathematics.

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