Transforming Log & Exponential Equations

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The discussion focuses on converting logarithmic equations to exponential form and vice versa, specifically addressing the equation ln 0.5 = -0.6931. Participants are encouraged to differentiate the function y = e^x(sin x + cos x) with respect to x. There is an emphasis on posting individual attempts to facilitate assistance from others. The thread highlights the collaborative nature of solving mathematical problems. Engaging in this process can enhance understanding of logarithmic and exponential relationships.
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Write the following log equations as exponential equations and vice-versa.
1.) ln 0.5 = - 0.6931

Differentiate with respect to x.
2.) y = e^x(sin x + cos x)
 
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Question: A clock's minute hand has length 4 and its hour hand has length 3. What is the distance between the tips at the moment when it is increasing most rapidly?(Putnam Exam Question) Answer: Making assumption that both the hands moves at constant angular velocities, the answer is ## \sqrt{7} .## But don't you think this assumption is somewhat doubtful and wrong?

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