If `log2=0.301,log3=0.477`, find the number of digits in `(108)^(10)`

Options

A. 21
B. 27
C. 20
D. 18

Answer

To find the number of digits in \( (108)^{10} \), we can use the formula for the number of digits of a number \( n \), which is given by:

\[ \text{Number of digits} = \lfloor \log_{10} n \rfloor + 1 \]

### Step 1: Express \( 108 \) in terms of its prime factors
First, we can express \( 108 \) as a product of its prime factors:
\[ 108 = 2^2 \times 3^3 \]

### Step 2: Calculate \( (108)^{10} \)
Now, we can write:
\[ (108)^{10} = (2^2 \times 3^3)^{10} = 2^{20} \times 3^{30} \]

### Step 3: Find \( \log_{10}((108)^{10}) \)
Using the properties of logarithms, we can find:
\[ \log_{10}((108)^{10}) = \log_{10}(2^{20} \times 3^{30}) = \log_{10}(2^{20}) + \log_{10}(3^{30}) \]

### Step 4: Apply the power rule of logarithms
Using the power rule of logarithms:
\[ \log_{10}(2^{20}) = 20 \cdot \log_{10}(2) \]
\[ \log_{10}(3^{30}) = 30 \cdot \log_{10}(3) \]

### Step 5: Substitute the given values of \( \log_{10}(2) \) and \( \log_{10}(3) \)
Given that \( \log_{10}(2) = 0.301 \) and \( \log_{10}(3) = 0.477 \):
\[ \log_{10}(2^{20}) = 20 \cdot 0.301 = 6.02 \]
\[ \log_{10}(3^{30}) = 30 \cdot 0.477 = 14.31 \]

### Step 6: Combine the logarithmic results
Now, we can add these two results together:
\[ \log_{10}((108)^{10}) = 6.02 + 14.31 = 20.33 \]

### Step 7: Find the number of digits
Finally, we apply the formula for the number of digits:
\[ \text{Number of digits} = \lfloor 20.33 \rfloor + 1 = 20 + 1 = 21 \]

### Final Answer
The number of digits in \( (108)^{10} \) is \( 21 \).

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