IMAT 2012 Q76 [Natural Logarithms]

Which of the following is equivalent to {ln(x^2y) - 2 ln(xy) + 3 ln y} ?

A. 0
B. { ln x + ln y}
C. { 2 ln y}
D. { 2ln x + 2 ln y}
E. { ln x + 2 ln y}

Using Natural Logarithm laws, we can conclude:

ln (x^2y) - 2 ln(xy) + 3 lny = ln {[\frac{x^2y}{(xy)^2}] y^3} = ln [(\frac{1}{y})y^3] = lny^2 = 2 lny

Leaving our answer to be C.

The Natural Logarithm laws:

  1. \log _{a}x\cdot y = \log _{a}x + \log _{a}y
  2. \log _{a}\frac{x}{y} = \log _{a}x - \log _{a}y
  3. \log _{a} x^{m} = m \log _{a}x
  4. \log _{a} \sqrt[n]{x} = \frac{1}{n} \log _{a}x

p.s. \ln =\log _{e}