Determine whether the following equation is true or false:

[tex]\[ \ln (x+y) = \ln x + \ln y \][/tex]

A. True
B. False



Answer :

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.