If `log_10 7=0.8451,` find the position of the first significant figure in `7^-20.`

Options

A. 15
B. 20
C. 17
D. 18

Explanation

To find the position of the first significant figure in \( 7^{-20} \), we will follow these steps:

### Step 1: Define the variable
Let \( x = 7^{-20} \).

### Step 2: Take the logarithm
We take the logarithm (base 10) of both sides:
\[ \log_{10} x = \log_{10} (7^{-20}) \]

### Step 3: Apply the logarithmic property
Using the property of logarithms that states \( \log_{10} (a^n) = n \cdot \log_{10} a \), we can rewrite the equation:
\[ \log_{10} x = -20 \cdot \log_{10} 7 \]

### Step 4: Substitute the given value
We know from the problem that \( \log_{10} 7 = 0.8451 \). Substituting this value in, we have:
\[ \log_{10} x = -20 \cdot 0.8451 \]

### Step 5: Calculate the logarithm
Now we perform the multiplication:
\[ \log_{10} x = -16.902 \]

### Step 6: Rewrite the logarithm
We can express this logarithmic value in a different form:
\[ \log_{10} x = -17 + 1 - 0.902 \]
This simplifies to:
\[ \log_{10} x = -16 + 0.098 \]

### Step 7: Find the position of the first significant figure
The logarithm \( \log_{10} x = -16.902 \) indicates that \( x \) is a number between \( 10^{-17} \) and \( 10^{-16} \). The first significant figure will be the first non-zero digit after the decimal point in the decimal representation of \( x \).

Since \( -16.902 \) indicates that the first significant figure is at position 17, we conclude that:
\[ \text{The position of the first significant figure in } 7^{-20} \text{ is } 17. \]

### Final Answer
The position of the first significant figure is **17**.

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