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Question

express the given quanity as a single logarithim

ln x +(a ln y) - (b ln z)

Answer 1

ln x +(a ln y) - (b ln z)
Since
ln (a b)=ln(a) + ln(b)
and
ln(x^a)= a ln(x)

We have

ln(x)+ ln(y^a) – (ln(z^b)=

=ln((x y^a )/z^b)

Answer 2

ln is just log with a subscript of e

Use these properties to help you in your problems in logarithms.
(I have restated these properties with ln for the convenience of this problem)

(1) ln(ac)=ln(a)+ln(c)
(2) ln(a/c)=ln(a)-ln(c)
(3) ln(a^R)= Rln(a)
(4) ln(1/c)=-ln (c)

ln(x) + Aln(y) -Bln(z)

Using (3) on the 2nd and 3rd terms we arrive at:

ln(x) + ln(y^a) -ln(z^B)

Using (1) on the 1rst and 2nd terms:

ln(x*y^a) - ln(z^B)

Using (2) on the remaining two terms

ln((x*y^a)/ (z^b))

Forgive my use of capitals. It doesn't look right to me while I'm doing the problem.

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Answer 3

ln x + (a ln y) - (b ln z) =
ln x + ln(y^a) - ln(z^b) =
ln((xy^a)/(z^b))

Answer 4

ln[xy^a/z^b]