What is the number of digits in the numeral form of `8^(17)?
( Given log_(10)^(2)==0.3010)`

Options

A. 51
B. 16
C. 15
D. 14

Explanation

To find the number of digits in the numeral form of \(8^{17}\), we can follow these steps:

### Step 1: Express \(8^{17}\) in terms of base 2
We know that \(8\) can be expressed as \(2^3\). Therefore:
\[ 8^{17} = (2^3)^{17} = 2^{51} \]

### Step 2: Set \(x\) equal to \(8^{17}\)
Let:
\[ x = 8^{17} = 2^{51} \]

### Step 3: Take the logarithm (base 10) of both sides
Taking logarithm base 10 on both sides gives:
\[ \log_{10} x = \log_{10} (2^{51}) \]

### Step 4: Use the logarithmic property
Using the property of logarithms that states \(\log_b (a^c) = c \cdot \log_b a\), we can rewrite the equation as:
\[ \log_{10} x = 51 \cdot \log_{10} 2 \]

### Step 5: Substitute the value of \(\log_{10} 2\)
We are given that \(\log_{10} 2 = 0.3010\). Substituting this value in gives:
\[ \log_{10} x = 51 \cdot 0.3010 \]

### Step 6: Calculate the value
Now, we calculate:
\[ \log_{10} x = 51 \cdot 0.3010 = 15.351 \]

### Step 7: Determine the number of digits
The number of digits \(d\) in a number \(x\) can be found using the formula:
\[ d = \lfloor \log_{10} x \rfloor + 1 \]
Here, \(\lfloor 15.351 \rfloor\) is \(15\). Therefore:
\[ d = 15 + 1 = 16 \]

### Final Answer
The number of digits in the numeral form of \(8^{17}\) is \(16\).

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