What is the value of `(log " "_(sqrt(alphabeta))H)/(log " "_(sqrt(alpha beta gamma))H)` ?

Options

A. `log_(alphabeta)(alpha)`
B. `log_(alphabetagamma)(alphabeta)`
C. `log_(alphabeta)(alphabetagamma)`
D. `log_(alphabeta)(beta)`

Answer

To solve the problem, we need to find the value of the expression:

\[ \frac{\log_{\sqrt{\alpha \beta}} H}{\log_{\sqrt{\alpha \beta \gamma}} H} \]

### Step 1: Apply the Change of Base Formula
Using the change of base formula for logarithms, we can rewrite the expression as:

\[ \frac{\log H / \log \sqrt{\alpha \beta}}{\log H / \log \sqrt{\alpha \beta \gamma}} \]

### Step 2: Simplify the Expression
This simplifies to:

\[ \frac{\log H}{\log \sqrt{\alpha \beta}} \cdot \frac{\log \sqrt{\alpha \beta \gamma}}{\log H} \]

The \(\log H\) terms cancel out, leaving us with:

\[ \frac{\log \sqrt{\alpha \beta \gamma}}{\log \sqrt{\alpha \beta}} \]

### Step 3: Use the Property of Logarithms
Now, we can use the property of logarithms that states \(\log a^b = b \cdot \log a\). Thus, we can rewrite the logarithms:

\[ \frac{\frac{1}{2} \log (\alpha \beta \gamma)}{\frac{1}{2} \log (\alpha \beta)} \]

### Step 4: Cancel Out the Common Factor
The \(\frac{1}{2}\) cancels out:

\[ \frac{\log (\alpha \beta \gamma)}{\log (\alpha \beta)} \]

### Step 5: Apply the Logarithm Property
Using the property of logarithms that states \(\log a - \log b = \log \frac{a}{b}\), we can express this as:

\[ \log_{\alpha \beta} (\alpha \beta \gamma) \]

### Final Result
Thus, the final value of the expression is:

\[ \log_{\alpha \beta} (\alpha \beta \gamma) \]