Expert: Ahmed Salami - 2/13/2006

Question
1. State the domain:  g(x)= 2 + ln x

2. Write the expression as a log of a single quantity: 3 ln x + 2 ln y - 4 ln z


Answer
Hi Oz,
Sorry for the time it took.
For f(y) = lny
The domain is y > 0, because the logarithm of a negative
number does not exist.
Now, for g(x) = 2 + lnx
g(x) = ln(e^2) + lnx
g(x) = ln(x.e^2)
The domain is therefore x.e^2 > 0
or simply x > 0
You could well have determined this from just the domain
of lnx.

From the laws of logarithm,
nlogx = log(x^n)
So,
3lnx + 2lny - 4lnz
= ln(x^3) + ln(y^2) - ln(z^4)
Also,
logA + logB = log(AB)
which implies
ln(x^3) + ln(y^2) - ln(z^4)
= ln[(x^3)(y^2)] - ln(z^4)
And,
logA - logB = log(A/B)
which gives
ln[(x^3)(y^2)] - ln(z^4)
= ln[(x^3)(y^2)/(z^4)]
It might be late but i sure do owe you the solution.
I hope i have helped. You can always get back to me.
Regards.