If `log 3=0.477` and `(1000)^x=3` , then `x` is equal to

Options

A. 0.159
B. 10
C. 0.0477
D. 0.0159

Answer

To solve the equation \( (1000)^x = 3 \) given that \( \log 3 = 0.477 \), we can follow these steps:

### Step 1: Rewrite the equation in logarithmic form
We start with the equation:
\[ (1000)^x = 3 \]
Taking the logarithm of both sides, we have:
\[ \log((1000)^x) = \log(3) \]

### Step 2: Apply the power rule of logarithms
Using the power rule of logarithms, which states that \( \log(a^b) = b \cdot \log(a) \), we can rewrite the left side:
\[ x \cdot \log(1000) = \log(3) \]

### Step 3: Calculate \( \log(1000) \)
Since \( 1000 = 10^3 \), we can find \( \log(1000) \):
\[ \log(1000) = \log(10^3) = 3 \cdot \log(10) \]
Knowing that \( \log(10) = 1 \), we have:
\[ \log(1000) = 3 \cdot 1 = 3 \]

### Step 4: Substitute \( \log(1000) \) back into the equation
Now we substitute \( \log(1000) \) back into our equation:
\[ x \cdot 3 = \log(3) \]

### Step 5: Solve for \( x \)
Now we can solve for \( x \):
\[ x = \frac{\log(3)}{3} \]
Substituting the given value of \( \log(3) = 0.477 \):
\[ x = \frac{0.477}{3} \]

### Step 6: Calculate the value of \( x \)
Now we perform the division:
\[ x = 0.159 \]

### Final Answer
Thus, the value of \( x \) is:
\[ \boxed{0.159} \]

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