How many zeroes are three between the decimal point and the first significant figure in `(0.5)^(100)`? Given, that `log 1/2 = bar1.6990`

Answer

To find how many zeros are there between the decimal point and the first significant figure in \( (0.5)^{100} \), we can follow these steps:

### Step 1: Define the expression
Let \( y = (0.5)^{100} \).

### Step 2: Take the logarithm of both sides
Taking the logarithm of both sides gives us:
\[ \log y = \log((0.5)^{100}) \]

### Step 3: Use the power rule of logarithms
Using the power rule of logarithms, we can bring down the exponent:
\[ \log y = 100 \cdot \log(0.5) \]

### Step 4: Substitute the value of \( \log(0.5) \)
We know from the problem statement that \( \log(0.5) = -1.6990 \). Therefore:
\[ \log y = 100 \cdot (-1.6990) \]

### Step 5: Calculate \( \log y \)
Now, we calculate:
\[ \log y = -169.90 \]

### Step 6: Interpret the logarithmic result
The logarithm tells us the power of 10 that \( y \) is equal to. Since \( \log y = -169.90 \), this means:
\[ y = 10^{-169.90} \]

### Step 7: Determine the number of zeros
The expression \( 10^{-169.90} \) can be rewritten as:
\[ y = 0.000\ldots001 \quad (\text{with } 169 \text{ zeros followed by a significant figure}) \]
The first significant figure appears after 169 zeros.

### Conclusion
Thus, there are **169 zeros** between the decimal point and the first significant figure in \( (0.5)^{100} \).

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