Calculate: `log_(2)(2//3)+log_(4)(9//4)`

Options

A. 1
B. 2
C. 0
D. `3//2`

Answer

To solve the expression \( \log_{2}\left(\frac{2}{3}\right) + \log_{4}\left(\frac{9}{4}\right) \), we can follow these steps:

### Step 1: Rewrite the logarithm with base 4
We know that \( \log_{4}(x) \) can be rewritten using the change of base formula:
\[ \log_{4}(x) = \frac{\log_{2}(x)}{\log_{2}(4)} \]
Since \( \log_{2}(4) = 2 \), we can rewrite \( \log_{4}\left(\frac{9}{4}\right) \) as:
\[ \log_{4}\left(\frac{9}{4}\right) = \frac{\log_{2}\left(\frac{9}{4}\right)}{2} \]

### Step 2: Substitute back into the expression
Now we can substitute this back into our original expression:
\[ \log_{2}\left(\frac{2}{3}\right) + \log_{4}\left(\frac{9}{4}\right) = \log_{2}\left(\frac{2}{3}\right) + \frac{1}{2} \log_{2}\left(\frac{9}{4}\right) \]

### Step 3: Combine the logarithms
We can combine the logarithms using the property \( \log_{a}(m) + \log_{a}(n) = \log_{a}(m \cdot n) \):
\[ \log_{2}\left(\frac{2}{3}\right) + \frac{1}{2} \log_{2}\left(\frac{9}{4}\right) = \log_{2}\left(\frac{2}{3}\right) + \log_{2}\left(\left(\frac{9}{4}\right)^{\frac{1}{2}}\right) \]
This simplifies to:
\[ \log_{2}\left(\frac{2}{3} \cdot \sqrt{\frac{9}{4}}\right) \]

### Step 4: Simplify the argument
Now we simplify the argument:
\[ \sqrt{\frac{9}{4}} = \frac{3}{2} \]
Thus, we have:
\[ \frac{2}{3} \cdot \frac{3}{2} = 1 \]

### Step 5: Evaluate the logarithm
Now we can evaluate the logarithm:
\[ \log_{2}(1) = 0 \]

### Final Answer
Thus, the final answer is:
\[ \log_{2}\left(\frac{2}{3}\right) + \log_{4}\left(\frac{9}{4}\right) = 0 \]