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 solve the problem, we need to find the number of zeros between the decimal point and the first significant figure of \(2^{-34}\) given that \(\log 2 = 0.301\).

### Step-by-Step Solution:

1. **Define the variable**: Let \(x = 2^{-34}\).

2. **Take the logarithm**: We will take the logarithm of both sides.
\[ \log x = \log(2^{-34}) \]

3. **Use the power rule of logarithms**: According to the power rule, \(\log(a^b) = b \cdot \log a\).
\[ \log x = -34 \cdot \log 2 \]

4. **Substitute the value of \(\log 2\)**: We know that \(\log 2 = 0.301\).
\[ \log x = -34 \cdot 0.301 \]

5. **Calculate \(\log x\)**:
\[ \log x = -10.234 \]

6. **Convert logarithmic form to exponential form**: If \(\log x = n\), then \(x = 10^n\).
\[ x = 10^{-10.234} \]

7. **Interpret the result**: The expression \(10^{-10.234}\) indicates that \(x\) is a very small number. Specifically, it can be expressed as:
\[ x = \frac{1}{10^{10.234}} \]

8. **Determine the number of zeros**: The power of \(10\) tells us how many zeros are between the decimal point and the first significant figure. Since \(10^{-10.234}\) means that there are 10 zeros after the decimal point before reaching the first significant figure.

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