Express `"log"(3sqrt(a^(2)))/(b^(5)sqrt(x))" or "(a^(2//3))/(b^(5)sqrt(c))` in terms of log a, log b and log c.

Options

A. `3/2loga+5logb-2logc`
B. `2/3loga-5logb-1/2logc`
C. `2/3loga-5logb+1/2logc`
D. `3/2loga+5logb-1/2logc`

Answer

To express the logarithmic expressions \(\log\left(\frac{3\sqrt{a^2}}{b^5\sqrt{x}}\right)\) and \(\log\left(\frac{a^{2/3}}{b^5\sqrt{c}}\right)\) in terms of \(\log a\), \(\log b\), and \(\log c\), we will follow these steps:

### Step 1: Simplify the Logarithmic Expression
We start with the expression:
\[ \log\left(\frac{3\sqrt{a^2}}{b^5\sqrt{x}}\right) \]

### Step 2: Apply the Quotient Rule
Using the logarithmic property \(\log\left(\frac{A}{B}\right) = \log A - \log B\), we can separate the logarithm:
\[ \log(3\sqrt{a^2}) - \log(b^5\sqrt{x}) \]

### Step 3: Simplify Each Logarithm
Now, we simplify each part:
1. For \(\log(3\sqrt{a^2})\):
- We can express \(\sqrt{a^2}\) as \(a\), so:
\[ \log(3a) = \log 3 + \log a \]
2. For \(\log(b^5\sqrt{x})\):
- We can express \(\sqrt{x}\) as \(x^{1/2}\), so:
\[ \log(b^5\sqrt{x}) = \log(b^5) + \log(\sqrt{x}) = 5\log b + \frac{1}{2}\log x \]

### Step 4: Combine the Results
Now we combine the results:
\[ \log(3\sqrt{a^2}) - \log(b^5\sqrt{x}) = (\log 3 + \log a) - (5\log b + \frac{1}{2}\log x) \]
This simplifies to:
\[ \log 3 + \log a - 5\log b - \frac{1}{2}\log x \]

### Step 5: Final Expression
Thus, the expression \(\log\left(\frac{3\sqrt{a^2}}{b^5\sqrt{x}}\right)\) can be expressed as:
\[ \log 3 + \log a - 5\log b - \frac{1}{2}\log x \]