What is the value of x in the following expansion ?
`1-log_(10)5=1/3("log"_(10)1/2+log_(10)x+1/3log_(10)5)`

Options

A. 1
B. `16xx5^(-1//3)`
C. `16xx5^(1//3)`
D. None of these

Answer

To solve the equation \( 1 - \log_{10} 5 = \frac{1}{3} \left( \log_{10} \frac{1}{2} + \log_{10} x + \frac{1}{3} \log_{10} 5 \right) \), we will follow these steps:

### Step 1: Rewrite the Left Side
We can rewrite the left side using the property of logarithms:
\[ 1 = \log_{10} 10 \quad \text{(since } \log_{10} 10 = 1\text{)} \]
Thus, we can express the left side as:
\[ \log_{10} 10 - \log_{10} 5 = \log_{10} \left( \frac{10}{5} \right) = \log_{10} 2 \]

### Step 2: Rewrite the Right Side
Now, let's simplify the right side:
\[ \frac{1}{3} \left( \log_{10} \frac{1}{2} + \log_{10} x + \frac{1}{3} \log_{10} 5 \right) \]
Using the property of logarithms \( \log a + \log b = \log(ab) \):
\[ \log_{10} \frac{1}{2} + \log_{10} x = \log_{10} \left( \frac{x}{2} \right) \]
Thus, we can rewrite the right side as:
\[ \frac{1}{3} \left( \log_{10} \left( \frac{x}{2} \right) + \frac{1}{3} \log_{10} 5 \right) \]

### Step 3: Combine the Logarithms
Now we can combine the logarithms:
\[ \frac{1}{3} \left( \log_{10} \left( \frac{x}{2} \right) + \log_{10} 5^{1/3} \right) = \frac{1}{3} \log_{10} \left( \frac{x}{2} \cdot 5^{1/3} \right) \]

### Step 4: Set the Two Sides Equal
Now we have:
\[ \log_{10} 2 = \frac{1}{3} \log_{10} \left( \frac{x}{2} \cdot 5^{1/3} \right) \]

### Step 5: Eliminate the Fraction
To eliminate the fraction, we multiply both sides by 3:
\[ 3 \log_{10} 2 = \log_{10} \left( \frac{x}{2} \cdot 5^{1/3} \right) \]

### Step 6: Rewrite the Left Side
Using the property \( n \log a = \log a^n \):
\[ \log_{10} (2^3) = \log_{10} \left( \frac{x}{2} \cdot 5^{1/3} \right) \]
This simplifies to:
\[ \log_{10} 8 = \log_{10} \left( \frac{x}{2} \cdot 5^{1/3} \right) \]

### Step 7: Set the Arguments Equal
Since the logarithms are equal, we can set the arguments equal:
\[ 8 = \frac{x}{2} \cdot 5^{1/3} \]

### Step 8: Solve for x
Multiply both sides by 2:
\[ 16 = x \cdot 5^{1/3} \]
Now, divide both sides by \( 5^{1/3} \):
\[ x = \frac{16}{5^{1/3}} \]

### Final Answer
Thus, the value of \( x \) is:
\[ x = 16 \cdot 5^{-1/3} \]

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