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The solution to the equation \( 12 \cdot 5^{x-1} + 3 \cdot 5^x - 5^{x-1} = 10 \) is:

\[
x = 2 - \frac{\log(13)}{\log(5)}
\]

This is the exact solution expressed in logarithmic form. If you'd like a numerical approximation or more details on how this solution was derived, feel free to ask!

Do you have any questions or would you like to explore this further?

Here are 5 related questions to expand on this:

1. How can I simplify exponential equations further before solving them?
2. What is the numerical approximation of \( x \) for this solution?
3. How does changing the base of an exponential function affect its solution?
4. Can we generalize this solution for other bases, like base 2 or 10?
5. What are the rules for solving equations involving logarithms?

**Tip:** When solving exponential equations, isolating the term with the exponent first can help simplify the process.

Learn how to solve the exponential equation 12 * 5^(x-1) + 3 * 5^x - 5^(x-1) = 10 with a detailed step-by-step guide. This problem explores algebraic concepts such as exponents and logarithms, suitable for Grades 10-12.

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