how do i rewrite this as a single logarithm?

Question

2ln3+(1/2)ln4

can someone please not only give me an answer, but explain step by step how they got this? thanks

Answer 1

Numbers multiplied by logs used to be exponents, and logs that are added use to be factors (multiplied).
So 2 ln 3 = ln (3^2); 1/2 ln 4 = ln 4^(1/2), and the sum changes to
ln[(3^2)(4^(1/2))]

Answer 2

Coefficient of logs are transforms of EXPONENTS of the arguement of the logs. So, we can rewrite this as
ln 9 + ln 2. Since the addition of logs is a transform of the multiplication of arguements, the answer is ln 18.

Answer 3

okay, numbers in front of the ln means it is a power of the number after the ln, so you can make it ln3^2 and ln4^(1/2).
Then, ln(xy) is the same as ln(x) + ln(y), so by doing that in reverse you see that ln{3^2*4^(1/2)} is the answer. i think.

Answer 4

ln(A) + ln(B) = ln(A*B)
n*ln(A) = ln(A^n)