If ` log_(10)2= 0.3010`, then the number of digits in `16^(12)` is

Options

A. 14
B. 15
C. 13
D. 16

Explanation

To find the number of digits in \( 16^{12} \), we can use the formula for the number of digits \( d \) of a number \( n \), which is given by:

\[ d = \lfloor \log_{10} n \rfloor + 1 \]

In this case, \( n = 16^{12} \). Therefore, we need to calculate \( \log_{10} (16^{12}) \).

### Step 1: Use the power rule of logarithms
Using the power rule of logarithms, we can simplify \( \log_{10} (16^{12}) \):

\[ \log_{10} (16^{12}) = 12 \cdot \log_{10} (16) \]

### Step 2: Express 16 in terms of powers of 2
Next, we express \( 16 \) as a power of \( 2 \):

\[ 16 = 2^4 \]

### Step 3: Substitute into the logarithm
Now we can substitute this back into our logarithm:

\[ \log_{10} (16) = \log_{10} (2^4) = 4 \cdot \log_{10} (2) \]

### Step 4: Substitute the value of \( \log_{10} (2) \)
We know from the problem statement that \( \log_{10} (2) = 0.3010 \). Therefore:

\[ \log_{10} (16) = 4 \cdot 0.3010 = 1.204 \]

### Step 5: Calculate \( \log_{10} (16^{12}) \)
Now we can substitute this back into our earlier equation:

\[ \log_{10} (16^{12}) = 12 \cdot 1.204 = 14.448 \]

### Step 6: Find the number of digits
Now we can find the number of digits \( d \):

\[ d = \lfloor 14.448 \rfloor + 1 = 14 + 1 = 15 \]

Thus, the number of digits in \( 16^{12} \) is **15**.

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