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The equation given is:

\[
4^{5x - 2} = 3^{-2x + 1}
\]

You are required to solve for \(x\) and express the answer in the form \( x = \frac{\ln a}{\ln b} \).

### Step 1: Apply the natural logarithm to both sides
We can take the natural logarithm (\(\ln\)) of both sides to deal with the exponents.

\[
\ln(4^{5x - 2}) = \ln(3^{-2x + 1})
\]

### Step 2: Use the logarithmic power rule
The logarithmic power rule states that \(\ln(a^b) = b \cdot \ln(a)\). Apply this rule to both sides:

\[
(5x - 2) \cdot \ln(4) = (-2x + 1) \cdot \ln(3)
\]

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

\[
(5x) \cdot \ln(4) - 2 \cdot \ln(4) = (-2x) \cdot \ln(3) + \ln(3)
\]

### Step 4: Group like terms
Now, group all the terms involving \(x\) on one side and constants on the other side.

\[
(5x) \cdot \ln(4) + (2x) \cdot \ln(3) = \ln(3) + 2 \cdot \ln(4)
\]

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

\[
x \left( 5 \cdot \ln(4) + 2 \cdot \ln(3) \right) = \ln(3) + 2 \cdot \ln(4)
\]

### Step 6: Solve for \(x\)
Now, solve for \(x\) by dividing both sides by \((5 \cdot \ln(4) + 2 \cdot \ln(3))\):

\[
x = \frac{\ln(3) + 2 \cdot \ln(4)}{5 \cdot \ln(4) + 2 \cdot \ln(3)}
\]

### Final Answer:
This is the value of \(x\) in the required form:

\[
x = \frac{\ln(3) + 2 \cdot \ln(4)}{5 \cdot \ln(4) + 2 \cdot \ln(3)}
\]

Let me know if you need further clarification or details!

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Would you like to explore more details on this solution? Here are 5 related questions:
1. How do logarithmic rules apply when solving exponential equations?
2. Why do we use natural logarithms specifically in this case?
3. Can the solution be simplified further or approximated?
4. How would the solution differ if the bases were the same on both sides of the equation?
5. What real-world problems involve solving equations of this form?

**Tip**: When solving logarithmic equations, always ensure the base is valid (i.e., greater than zero and not equal to one) for the natural logarithm function.

Solve 4^(5x-2) = 3^(-2x+1), giving the answer in the form x = ln(a)/ln(b)

Learn how to solve the exponential equation 4^(5x-2) = 3^(-2x+1) using logarithmic rules. This solution demonstrates step-by-step methods to express x in the form ln(a)/ln(b), suitable for students in Grades 10-12.

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