If log2=0.301, the number of zeroes between the decimal point and the first significant figure of `2^(-34)` is

Options

A. 9
B. 10
C. 11
D. 12

Answer

To find the number of zeros between the decimal point and the first significant figure of \(2^{-34}\), we will follow these steps:

### Step 1: Define the expression
Let \(X = 2^{-34}\).

### Step 2: Take the logarithm
We take the logarithm of both sides:
\[ \log X = \log(2^{-34}) \]

### Step 3: Apply the power rule of logarithms
Using the power rule of logarithms, we can rewrite the expression:
\[ \log X = -34 \log 2 \]

### Step 4: Substitute the value of \(\log 2\)
Given that \(\log 2 = 0.301\), we substitute this value into the equation:
\[ \log X = -34 \times 0.301 \]

### Step 5: Calculate \(-34 \times 0.301\)
Now, we perform the multiplication:
\[ \log X = -10.234 \]

### Step 6: Convert logarithm back to exponential form
To find \(X\), we convert back from logarithmic form:
\[ X = 10^{-10.234} \]

### Step 7: Analyze the number of zeros
The expression \(10^{-10.234}\) can be interpreted as:
\[ X = \frac{1}{10^{10.234}} = \frac{1}{10^{10} \times 10^{0.234}} \]
This indicates that \(X\) is a very small number.

### Step 8: Determine the number of zeros
The number \(10^{-10.234}\) means there are 10 zeros after the decimal point before reaching the first significant figure. The first significant figure will appear after these 10 zeros.

### Conclusion
Thus, the number of zeros between the decimal point and the first significant figure of \(2^{-34}\) is **10**.

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