MHB How can the properties of logarithms be used to simplify and solve equations?

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The discussion focuses on using properties of logarithms to simplify expressions and solve equations. Participants are encouraged to express logarithmic equations as sums, differences, or multiples of logarithms. One user successfully simplified an expression to "lnx - 1/2ln(x^2+1)" but struggles with another problem involving a denominator and a product in the numerator. The conversation emphasizes the need for clarity on handling logarithmic properties in various scenarios. Overall, the thread highlights common challenges and encourages collaborative problem-solving in logarithmic simplification.
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Use the properties of Logarithms to write the expression as a sum, difference, and/or constant multiple of logarithms:
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What have you tried? Where are you stuck?
 
Ackbach said:
What have you tried? Where are you stuck?

For the first one I did " lnx-1/2ln(x^2+1)"

For the second one, I have no idea what to do :(
 
What should you do with the denominator? How about the product in the numerator?
 

Thread 'Area of Overlapping Squares'

Here is a little puzzle from the book 100 Geometric Games by Pierre Berloquin. The side of a small square is one meter long and the side of a larger square one and a half meters long. One vertex of the large square is at the center of the small square. The side of the large square cuts two sides of the small square into one- third parts and two-thirds parts. What is the area where the squares overlap?
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