We are given the system of equations:

\[
652 = a \cdot \ln(500) + b
\]
\[
648 = a \cdot \ln(100) + b
\]

We can solve this system to find the values of \(a\) and \(b\). 

### Steps to solve:
1. Subtract the second equation from the first to eliminate \(b\):
   \[
   (652 - 648) = a \cdot (\ln(500) - \ln(100))
   \]
2. Simplify this equation to find \(a\), then substitute the value of \(a\) back into one of the original equations to find \(b\).

Let's compute this step-by-step.The solution to the system of equations is:

\[
a \approx 2.4853
\]
\[
b \approx 636.5546
\]

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

Here are 5 related questions to consider:
1. How can we verify the solution by substituting the values of \(a\) and \(b\) back into the original equations?
2. What is the significance of the natural logarithm in this context?
3. Can this system of equations be solved graphically?
4. How would the process change if the logarithms had a different base?
5. Can we generalize this method to solve other linear systems with logarithmic terms?

**Tip:** When dealing with logarithmic equations, make sure the arguments of the logarithms are positive, as the logarithm function is undefined for non-positive values.

652 = a⋅ln(500) + b
648 = a⋅ln(100) + b

Explore how to solve the system of equations 652 = a⋅ln(500) + b and 648 = a⋅ln(100) + b using logarithmic properties and algebraic manipulation. The step-by-step solution helps you understand logarithmic equations and systems of linear equations, ideal for high school students in Grades 10-12.

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