What is the value of `(log_(sqrt(alphabeta))(H))/(log_(alphabeta)(beta))`

Options

A. `log_(alphabeta)(H^2)`
B. `log_(alpha)(H)`
C. `log_(alphabeta)(H)`
D. `log_(beta)(H^2)`

Answer

To solve the expression \((\log_{\sqrt{\alpha\beta}}(H))/(\log_{\alpha\beta}(\beta))\), we will use properties of logarithms step by step.

### Step 1: Rewrite the logarithms using the change of base formula
Using the change of base formula, we can express the logarithms in terms of natural logarithms (or any other base). The change of base formula states that:
\[ \log_b(a) = \frac{\log_k(a)}{\log_k(b)} \]
Applying this to our expression, we have:
\[ \log_{\sqrt{\alpha\beta}}(H) = \frac{\log(H)}{\log(\sqrt{\alpha\beta})} \]
and
\[ \log_{\alpha\beta}(\beta) = \frac{\log(\beta)}{\log(\alpha\beta)} \]

### Step 2: Substitute these into the original expression
Now substituting these into our original expression, we get:
\[ \frac{\log(H)/\log(\sqrt{\alpha\beta})}{\log(\beta)/\log(\alpha\beta)} \]

### Step 3: Simplify the expression
This simplifies to:
\[ \frac{\log(H)}{\log(\sqrt{\alpha\beta})} \cdot \frac{\log(\alpha\beta)}{\log(\beta)} \]

### Step 4: Simplify \(\log(\sqrt{\alpha\beta})\)
Recall that \(\sqrt{\alpha\beta} = (\alpha\beta)^{1/2}\). Therefore,
\[ \log(\sqrt{\alpha\beta}) = \log((\alpha\beta)^{1/2}) = \frac{1}{2} \log(\alpha\beta) \]

### Step 5: Substitute back into the expression
Now substituting this back into our expression gives:
\[ \frac{\log(H)}{\frac{1}{2} \log(\alpha\beta)} \cdot \frac{\log(\alpha\beta)}{\log(\beta)} \]

### Step 6: Cancel \(\log(\alpha\beta)\)
The \(\log(\alpha\beta)\) terms cancel out:
\[ \frac{\log(H)}{\frac{1}{2}} \cdot \frac{1}{\log(\beta)} = 2 \cdot \frac{\log(H)}{\log(\beta)} \]

### Step 7: Rewrite using the properties of logarithms
Using the property of logarithms, we can rewrite this as:
\[ 2 \log_{\beta}(H) \]

### Conclusion
Thus, the final value of the expression \((\log_{\sqrt{\alpha\beta}}(H))/(\log_{\alpha\beta}(\beta))\) is:
\[ \log_{\beta}(H^2) \]