Express ln(√𝑥𝑦 ) in terms of ln 𝑥 and ln 𝑦:

To express ln(√𝑥𝑦 ) in terms of ln 𝑥 and ln 𝑦, we can use the properties of logarithms. Here are the steps:

1. First, recognize that √𝑥𝑦 is the same as (𝑥𝑦)^(1/2). So, ln(√𝑥𝑦 ) becomes ln((𝑥𝑦)^(1/2)).

2. Use the power rule of logarithms, which states that ln(a^b) = b*ln(a). Applying this rule, ln((𝑥𝑦)^(1/2)) becomes (1/2)*ln(𝑥𝑦).

3. Now, use the product rule of logarithms, which states that ln(ab) = ln(a) + ln(b). Applying this rule, (1/2)ln(𝑥𝑦) becomes (1/2)(ln(𝑥) + ln(𝑦)).

4. Finally, distribute the 1/2 to both ln(𝑥) and ln(𝑦) to get (1/2)*ln(𝑥) + (1/2)*ln(𝑦).

So, ln(√𝑥𝑦 ) in terms of ln 𝑥 and ln 𝑦 is (1/2)*ln(𝑥) + (1/2)*ln(𝑦).