If `log_(e) x = s and log_(e) , y = t`, then `log_(e)(xy)^(2) = ?`

Options

A. `2(s + t)`
B. `s + t`
C. `4st`
D. `2st`

Answer

To solve the problem, we start with the given logarithmic equations:

1. We know that \( \log_e x = s \) and \( \log_e y = t \).

We need to find \( \log_e (xy)^2 \).

### Step 1: Rewrite the expression using properties of logarithms
Using the property of logarithms that states \( \log_b (m^n) = n \cdot \log_b m \), we can rewrite the expression:

\[ \log_e (xy)^2 = 2 \cdot \log_e (xy) \]

**Hint:** Remember that the logarithm of a power can be simplified by bringing the exponent in front.

### Step 2: Expand \( \log_e (xy) \)
Next, we can use the product property of logarithms, which states that \( \log_b (mn) = \log_b m + \log_b n \):

\[ \log_e (xy) = \log_e x + \log_e y \]

Substituting the values of \( \log_e x \) and \( \log_e y \):

\[ \log_e (xy) = s + t \]

**Hint:** When dealing with the logarithm of a product, you can separate it into the sum of the logarithms.

### Step 3: Substitute back into the expression
Now we substitute \( \log_e (xy) \) back into our expression from Step 1:

\[ \log_e (xy)^2 = 2 \cdot (s + t) \]

**Hint:** When you have a sum inside the logarithm, you can multiply it by the coefficient outside.

### Step 4: Final simplification
Distributing the 2 gives us:

\[ \log_e (xy)^2 = 2s + 2t \]

**Hint:** Always combine like terms when simplifying expressions.

### Final Answer
Thus, the final answer is:

\[ \log_e (xy)^2 = 2s + 2t \]

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