Calculating the Number of Digits Using Logarithms

To find the number of digits in a large number like \((108)^{10}\), we can use the concept of logarithms, specifically the base-10 logarithm. The number of digits in any positive integer \(N\) is given by the formula: Number of digits \( = \lfloor \log_{10} N \rfloor + 1 \). Here, \(\lfloor x \rfloor\) represents the greatest integer less than or equal to \(x\) (the floor function).

Our goal is to calculate \(\log_{10} (108)^{10}\) and then use its integer part to find the number of digits.

Applying Logarithm Properties

We need to evaluate \( \log_{10} (108)^{10} \). We can use the power property of logarithms, which states that \( \log_b (a^c) = c \log_b a \). Applying this, we get:

\( \log_{10} (108)^{10} = 10 \times \log_{10} 108 \)

Now we need to find the value of \( \log_{10} 108 \). We are given the values of \( \log_{10} 2 \) and \( \log_{10} 3 \). We can express 108 in terms of its prime factors 2 and 3:

\( 108 = 2 \times 54 \)
\( 108 = 2 \times 2 \times 27 \)
\( 108 = 2 \times 2 \times 3 \times 9 \)
\( 108 = 2 \times 2 \times 3 \times 3 \times 3 \)
\( 108 = 2^2 \times 3^3 \)

Now we can write \( \log_{10} 108 \) as \( \log_{10} (2^2 \times 3^3) \). Using the product property of logarithms, \( \log_b (xy) = \log_b x + \log_b y \), we get:

\( \log_{10} (2^2 \times 3^3) = \log_{10} (2^2) + \log_{10} (3^3) \)

Again, applying the power property of logarithms (\( \log_b (a^c) = c \log_b a \)) to each term:

\( \log_{10} (2^2) = 2 \log_{10} 2 \)
\( \log_{10} (3^3) = 3 \log_{10} 3 \)

So, \( \log_{10} 108 = 2 \log_{10} 2 + 3 \log_{10} 3 \).

Using the Given Logarithm Values

We are given that \( \log_{10} 2 = 0.301 \) and \( \log_{10} 3 = 0.477 \). Substitute these values:

\( \log_{10} 108 = 2 \times (0.301) + 3 \times (0.477) \)
\( \log_{10} 108 = 0.602 + 1.431 \)
\( \log_{10} 108 = 2.033 \)

Now we can go back to our original expression for \( \log_{10} (108)^{10} \):

\( \log_{10} (108)^{10} = 10 \times \log_{10} 108 \)
\( \log_{10} (108)^{10} = 10 \times (2.033) \)
\( \log_{10} (108)^{10} = 20.33 \)

Finding the Number of Digits

The value of \( \log_{10} (108)^{10} \) is 20.33. The integer part of this logarithm is the characteristic, which is 20.

The number of digits in a positive integer \(N\) is \( \lfloor \log_{10} N \rfloor + 1 \). In our case, \(N = (108)^{10}\), and \( \log_{10} N = 20.33 \).

Number of digits \( = \lfloor 20.33 \rfloor + 1 \)
Number of digits \( = 20 + 1 \)
Number of digits \( = 21 \)

Therefore, there are 21 digits in the number \((108)^{10}\).

Revision Table: Logarithm Calculation Steps

Additional Information on Logarithms and Number of Digits

The base-10 logarithm of a number helps us understand its magnitude. For any positive number \(N\), \( \log_{10} N = C + M \), where \(C\) is the integer part (called the characteristic) and \(M\) is the decimal part (called the mantissa, where \(0 \le M < 1\)).

• If \(N\) is an integer greater than or equal to 1, the number of digits in \(N\) is \( C + 1 \). This is because if \(N\) has \(k\) digits, it is between \(10^{k-1}\) and \(10^k\) (i.e., \(10^{k-1} \le N < 10^k\)). Taking the base-10 logarithm, we get \(k-1 \le \log_{10} N < k\). The characteristic \(C\) is thus \(k-1\), so the number of digits \(k = C+1\).
• In our calculation for \((108)^{10}\), we found \( \log_{10} (108)^{10} = 20.33 \). The characteristic is 20, and the mantissa is 0.33. According to the formula, the number of digits is \(20 + 1 = 21\).
• This method is extremely useful for determining the size of very large numbers without having to calculate the exact value.