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To determine if you simplified the expression correctly, let's go through the steps together.
Starting from the original expression: 3/2 ln25 + 3 ln2
1. Apply the power rule of logarithms to simplify ln2^3:
ln2^3 = ln(2*2*2) = ln8
2. Use the product rule of logarithms to combine ln125 and ln8:
ln125 + ln8 = ln(125*8)
3. Finally, apply the logarithmic rule to simplify ln(125*8):
ln(125*8) = ln(1000)
Therefore, your final answer should be ln1000. Based on the steps you provided, you have done it correctly. Well done!