Alright, let's address each of the expressions step-by-step:
(a) [tex]\(\ln(AB)\)[/tex]:
Using the logarithm property [tex]\(\ln(AB) = \ln(A) + \ln(B)\)[/tex], we substitute the given values [tex]\(x = \ln(A)\)[/tex] and [tex]\(y = \ln(B)\)[/tex]. So, we get:
[tex]\[
\ln(AB) = \ln(A) + \ln(B) = x + y
\][/tex]
(b) [tex]\(\ln\left(A^4 \cdot \sqrt{B}\right)\)[/tex]:
Using the properties of logarithms and exponents, we can break this down. First, [tex]\(A^4 \cdot \sqrt{B}\)[/tex] can be expressed as [tex]\(\ln(A^4) + \ln(\sqrt{B})\)[/tex]. Using the exponent rule [tex]\(\ln(A^n) = n \ln(A)\)[/tex], we have:
[tex]\[
\ln(A^4) = 4\ln(A) = 4x
\][/tex]
Next, [tex]\(\sqrt{B}\)[/tex] is [tex]\(B^{1/2}\)[/tex], so:
[tex]\[
\ln(\sqrt{B}) = \ln(B^{1/2}) = (1/2)\ln(B) = \frac{1}{2}y
\][/tex]
Combining these, we have:
[tex]\[
\ln\left(A^4 \cdot \sqrt{B}\right) = 4x + \frac{1}{2}y
\][/tex]
(c) [tex]\(\frac{\ln(A)}{\ln(B)}\)[/tex]:
We are given [tex]\(x = \ln(A)\)[/tex] and [tex]\(y = \ln(B)\)[/tex]. So, directly substituting, we get:
[tex]\[
\frac{\ln(A)}{\ln(B)} = \frac{x}{y}
\][/tex]
(d) [tex]\(\ln\left(\frac{A}{B}\right)\)[/tex]:
Using the logarithm property [tex]\(\ln\left(\frac{A}{B}\right) = \ln(A) - \ln(B)\)[/tex], we substitute [tex]\(x = \ln(A)\)[/tex] and [tex]\(y = \ln(B)\)[/tex] to get:
[tex]\[
\ln\left(\frac{A}{B}\right) = \ln(A) - \ln(B) = x - y
\][/tex]
(e) [tex]\(AB\)[/tex]:
Expressing [tex]\(AB\)[/tex] in terms of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] using exponentials, since [tex]\(AB = e^{\ln(AB)}\)[/tex], and [tex]\(\ln(AB) = x + y\)[/tex], we get:
[tex]\[
AB = e^{\ln(A) + \ln(B)} = e^{x + y}
\][/tex]
(f) [tex]\(\ln(A - B)\)[/tex]:
The expression [tex]\(\ln(A - B)\)[/tex] cannot be simplified directly in terms of [tex]\(\ln(A)\)[/tex] and [tex]\(\ln(B)\)[/tex] because logarithms do not have a simple property for the subtraction of numbers inside the logarithm. Therefore, [tex]\(\ln(A - B)\)[/tex] cannot be simplified using [tex]\(x\)[/tex] and [tex]\(y\)[/tex].
Recapping:
(a) [tex]\(\ln(AB) = x + y\)[/tex]
(b) [tex]\(\ln\left(A^4 \cdot \sqrt{B}\right) = 4x + (1/2)y\)[/tex]
(c) [tex]\(\frac{\ln(A)}{\ln(B)} = \frac{x}{y}\)[/tex]
(d) [tex]\(\ln\left(\frac{A}{B}\right) = x - y\)[/tex]
(e) [tex]\(AB = e^{x + y}\)[/tex]
(f) [tex]\(\ln(A - B)\)[/tex] cannot be simplified in terms of [tex]\(x\)[/tex] and [tex]\(y\)[/tex]
