If log2=0.301, the number of integers in the expansion of `4^(17)` is

Options

A. 9
B. 11
C. 13
D. 15

Answer

To solve the problem of finding the number of integers in the expansion of \(4^{17}\), we can follow these steps:

### Step 1: Express \(n\) in terms of logarithms
Let \(n = 4^{17}\). We want to find \(\log n\).

### Step 2: Use the logarithmic property
Using the property of logarithms that states \(\log(a^b) = b \cdot \log a\), we can write:
\[ \log n = \log(4^{17}) = 17 \cdot \log 4 \]

### Step 3: Rewrite \(\log 4\)
Since \(4\) can be expressed as \(2^2\), we have:
\[ \log 4 = \log(2^2) = 2 \cdot \log 2 \]

### Step 4: Substitute \(\log 4\) into the equation
Now substituting \(\log 4\) back into our expression for \(\log n\):
\[ \log n = 17 \cdot (2 \cdot \log 2) = 34 \cdot \log 2 \]

### Step 5: Substitute the given value of \(\log 2\)
We are given that \(\log 2 = 0.301\). Therefore:
\[ \log n = 34 \cdot 0.301 \]

### Step 6: Calculate \(\log n\)
Now we calculate:
\[ \log n = 34 \cdot 0.301 = 10.234 \]

### Step 7: Determine characteristics and mantissa
In logarithmic terms, \(\log n\) can be expressed as:
\[ \log n = 10 + 0.234 \]
Here, \(10\) is the characteristic and \(0.234\) is the mantissa.

### Step 8: Find the number of integers
The number of integers in the expansion of \(n\) is given by the characteristic plus one:
\[ \text{Number of integers} = 10 + 1 = 11 \]

### Final Answer
Thus, the number of integers in the expansion of \(4^{17}\) is \(11\).

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