Compute ` log_(ab)(root(3)a//sqrtb)" if " log_(ab) a = 4`.

Explanation

To compute \( \log_{ab}\left(\frac{\sqrt[3]{a}}{\sqrt{b}}\right) \) given that \( \log_{ab} a = 4 \), we can follow these steps:

### Step 1: Rewrite the logarithm using properties
We can use the property of logarithms that states:
\[ \log_b\left(\frac{x}{y}\right) = \log_b x - \log_b y \]
Thus, we can rewrite our expression as:
\[ \log_{ab}\left(\frac{\sqrt[3]{a}}{\sqrt{b}}\right) = \log_{ab}(\sqrt[3]{a}) - \log_{ab}(\sqrt{b}) \]

### Step 2: Simplify each logarithm
Next, we simplify each term:
- For \( \log_{ab}(\sqrt[3]{a}) \):
Using the property \( \log_b(x^n) = n \cdot \log_b(x) \), we have:
\[ \log_{ab}(\sqrt[3]{a}) = \log_{ab}(a^{1/3}) = \frac{1}{3} \log_{ab} a \]

- For \( \log_{ab}(\sqrt{b}) \):
Similarly, we have:
\[ \log_{ab}(\sqrt{b}) = \log_{ab}(b^{1/2}) = \frac{1}{2} \log_{ab} b \]

### Step 3: Substitute known values
Now we substitute the known value \( \log_{ab} a = 4 \) into our expression:
\[ \log_{ab}(\sqrt[3]{a}) = \frac{1}{3} \cdot 4 = \frac{4}{3} \]

Next, we need to find \( \log_{ab} b \). We can use the change of base formula:
\[ \log_{ab} b = \frac{\log b}{\log(ab)} = \frac{\log b}{\log a + \log b} \]

### Step 4: Find \( \log_{ab} b \)
Since we don't have a specific value for \( \log b \), we can denote it as \( x \):
\[ \log_{ab} b = \frac{x}{4 + x} \]

### Step 5: Combine the results
Now we can combine the results:
\[ \log_{ab}\left(\frac{\sqrt[3]{a}}{\sqrt{b}}\right) = \frac{4}{3} - \frac{1}{2} \cdot \frac{x}{4 + x} \]

### Step 6: Final expression
Thus, the final expression for \( \log_{ab}\left(\frac{\sqrt[3]{a}}{\sqrt{b}}\right) \) is:
\[ \log_{ab}\left(\frac{\sqrt[3]{a}}{\sqrt{b}}\right) = \frac{4}{3} - \frac{x}{2(4 + x)} \]