How do you condense lny+lntlny+lnt?

1 Answer
Aug 22, 2017

Provided y, t > 0y,t>0, we have:

ln y + ln t = ln (yt)lny+lnt=ln(yt)

Explanation:

Note that if a, b > 0a,b>0 then:

ln a + ln b = ln (ab)lna+lnb=ln(ab)

This follows from the corresponding property of exponents:

e^(p+q) = e^p * e^qep+q=epeq

since e^xex and ln xlnx are inverses of one another.

So given a, b > 0a,b>0, let:

p = ln a" "p=lna and " "q = ln b q=lnb.

Then:

e^p = a" "ep=a and " "e^q = b eq=b

So:

ln a + ln b = p + q = ln(e^(p+q)) = ln(e^p * e^q) = ln(ab)lna+lnb=p+q=ln(ep+q)=ln(epeq)=ln(ab)

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So provided y, t > 0y,t>0 we have:

ln y + ln t = ln (yt)lny+lnt=ln(yt)