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Natural Logarithm and Integral Properties

I was asked to prove that ln(xy) = ln x + ln y using the integral definition. While I'm not asking for any answers on the proof, I was wondering how to interpret and set-up this proof using the "integral definition" (As I am unsure what that means.)


And to prove that ln(x/y) = ln x - ln y

Is it right to say this?

$$\ln(\frac{x}{y})=\int_1^{\frac{x}{y}} \frac{dt}{t}=\int_1^x \frac{dt}{t}-\int_x^{\frac{x}{y}}\frac{dt}{t}.$$

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marked as duplicate by Sasha, William, Aang, Noah Snyder, J. M. Oct 5 '12 at 12:59

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

This wikipedia page might help you... – Envious Page Sep 16 '12 at 3:23
up vote 2 down vote accepted

By definition, $$\ln w=\int_1^w \frac{dt}{t}.$$ Thus $$\ln(xy)=\int_1^{xy} \frac{dt}{t}=\int_1^x \frac{dt}{t}+\int_x^{xy}\frac{dt}{t}.$$ Now make an appropriate change of variable to conclude that the last integral on the right is equal to $\ln y$.

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Thank you. Is this also the same technique I would use to prove that ln(x/y) = ln x - ln y? I edited my question to ask if that's right. – Johannes Sep 16 '12 at 4:03
@Johannes: No point in starting over. Note that $(x/y)(y)=x$. so by the above result $\ln(x)=\ln(x/y)+\ln(y)$. Basically finished! – André Nicolas Sep 16 '12 at 4:07

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