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    Using the definition of the natural log function (integral of 1/t dt from 1 to x), how would I prove ln(x) + ln(y) = ln(xy)
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    Note that \displaystyle \int_1^{xy} \dfrac{1}{t}\, dt = \int_1^{x} \dfrac{1}{t}\, dt + \int_x^{xy} \dfrac{1}{t}\, dt. Can you think of a useful substitution to make for the second integral? [Hint: compare its limits with the limits you want, namely 1 and y.]
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    Sorry, I'm being really dim today and can't spot it. I want x=1 for the limits on the integral furthest to the right so I get 1 and y?
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    (Original post by blap)
    Sorry, I'm being really dim today and can't spot it. I want x=1 for the limits on the integral furthest to the right so I get 1 and y?
    Put u = \dfrac{t}{x} (which is valid since x \ne 0).
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    Thanks, it's so obvious! I think I was getting confused with the x and y being together, and so not thinking to introduce some other variable.
 
 
 
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