Question

If log2 = 0.30103 and log3 = 0.4771, find the number of digits in (648)\(^5\).

• A. 23
• B. 15
• C. 13
• D. 14
• E. 22

Hint:

To find the number of digits, use the logarithmic properties and apply the formula \( D = \lfloor \log_{10} N \rfloor + 1 \).

Answer 1

The Correct Option is B

To find the number of digits in a number $N$, we use the formula: 
 
Number of digits = $\lfloor \log_{10} N \rfloor + 1$ 

We are given $N = (648)^5$ 

So, $\log_{10}((648)^5) = 5 \log_{10}(648)$ 

Factorize: $648 = 2^3 \cdot 3^4$ 
So, $\log_{10}(648) = \log_{10}(2^3 \cdot 3^4) = 3 \log_{10}2 + 4 \log_{10}3$ 
= $3 \times 0.30103 + 4 \times 0.4771 = 0.90309 + 1.9084 = 2.81149$ 

Then, $\log_{10}((648)^5) = 5 \times 2.81149 = 14.05745$ 

Number of digits = $\lfloor 14.05745 \rfloor + 1 = 14 + 1 = \mathbf{15}$ 

Answer: (b) 15

Answer 2

We know that the number of digits \( D \) in a number \( N \) is given by \( D = \lfloor \log_{10} N \rfloor + 1 \). We need to find the number of digits in \( (648)^5 \). \[ \log_{10} (648)^5 = 5 \log_{10} 648 \] First, calculate \( \log_{10} 648 \): \[ \log_{10} 648 = \log_{10} (6.48 \times 10^2) = \log_{10} 6.48 + 2 \] Using the values: \[ \log_{10} 6.48 = \log_{10} (2 \times 3.24) = \log_{10} 2 + \log_{10} 3.24 = 0.30103 + 0.5119 = 0.81293 \] Thus, \[ \log_{10} 648 = 0.81293 + 2 = 2.81293 \] Now, calculate \( 5 \times 2.81293 = 14.06465 \). The number of digits is \( \lfloor 14.06465 \rfloor + 1 = 15 \).