To determine whether the equation [tex]\(\ln (x+y)=\ln x+\ln y\)[/tex] is true, let's analyze the properties of logarithms.
One of the key properties of logarithms is the product rule, which states:
[tex]\[
\ln(xy) = \ln(x) + \ln(y)
\][/tex]
This means that the logarithm of a product is equal to the sum of the logarithms of the factors. However, this is different from the sum of logarithms. In the equation [tex]\(\ln (x+y)=\ln x+\ln y\)[/tex], we are dealing with the logarithm of a sum, not a product.
To understand why [tex]\(\ln (x+y)=\ln x+\ln y\)[/tex] is not generally true, let's consider a specific example. Suppose [tex]\(x = 2\)[/tex] and [tex]\(y = 3\)[/tex]:
[tex]\[
\ln (2+3) = \ln 5
\][/tex]
On the other hand:
[tex]\[
\ln 2 + \ln 3
\][/tex]
Evaluating these expressions, we get:
[tex]\[
\ln 5 \approx 1.6094
\][/tex]
And:
[tex]\[
\ln 2 \approx 0.6931
\][/tex]
[tex]\[
\ln 3 \approx 1.0986
\][/tex]
[tex]\[
\ln 2 + \ln 3 \approx 0.6931 + 1.0986 = 1.7917
\][/tex]
Clearly, [tex]\(\ln 5 \approx 1.6094\)[/tex] does not equal [tex]\(\ln 2 + \ln 3 \approx 1.7917\)[/tex]. This demonstrates that [tex]\(\ln(x+y)\)[/tex] is not the same as [tex]\(\ln(x) + \ln(y)\)[/tex].
Therefore, the statement [tex]\(\ln (x+y)=\ln x+\ln y\)[/tex] is False.
