Calculating Number of Digits Using Logarithms

To find the number of digits in a large number like $(54)^{10}$, we can use the concept of logarithms. The number of digits in any positive integer \(N\) is given by the formula: number of digits = \( \lfloor \log_{10} N \rfloor + 1 \).

Here, our number \(N = (54)^{10}\). We need to calculate \( \log_{10} (54)^{10} \).

Using the property of logarithms, \( \log_b (M^p) = p \log_b M \), we can write:

\( \log_{10} (54)^{10} = 10 \times \log_{10} 54 \)

Now, we need to calculate \( \log_{10} 54 \). We can break down 54 into its prime factors: \( 54 = 2 \times 27 = 2 \times 3^3 \).

Using the property of logarithms, \( \log_b (MN) = \log_b M + \log_b N \), we get:

\( \log_{10} 54 = \log_{10} (2 \times 3^3) = \log_{10} 2 + \log_{10} 3^3 \)

Using the property \( \log_b (M^p) = p \log_b M \) again for \( \log_{10} 3^3 \):

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

So, \( \log_{10} 54 = \log_{10} 2 + 3 \times \log_{10} 3 \)

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

Substitute these values into the equation for \( \log_{10} 54 \):

\( \log_{10} 54 = 0.301 + 3 \times 0.477 \)

First, calculate \( 3 \times 0.477 \):

\( 3 \times 0.477 = 1.431 \)

Now, calculate \( \log_{10} 54 \):

\( \log_{10} 54 = 0.301 + 1.431 = 1.732 \)

Now, we can find \( \log_{10} (54)^{10} \):

\( \log_{10} (54)^{10} = 10 \times \log_{10} 54 = 10 \times 1.732 = 17.32 \)

The number of digits in \( (54)^{10} \) is given by \( \lfloor \log_{10} (54)^{10} \rfloor + 1 \).

\( \lfloor 17.32 \rfloor = 17 \)

Number of digits = \( 17 + 1 = 18 \).

Thus, there are 18 digits in \( (54)^{10} \).

Summary of Calculation Steps

1. Identify the number $N = (54)^{10}$.
2. Recall the formula for the number of digits: $\lfloor \log_{10} N \rfloor + 1$.
3. Calculate $\log_{10} (54)^{10}$ using logarithm properties.
4. $\log_{10} (54)^{10} = 10 \times \log_{10} 54$.
5. Break down $\log_{10} 54$: $\log_{10} 54 = \log_{10} (2 \times 3^3) = \log_{10} 2 + 3 \log_{10} 3$.
6. Substitute given values: $\log_{10} 2 = 0.301$, $\log_{10} 3 = 0.477$.
7. Calculate: $\log_{10} 54 = 0.301 + 3 \times 0.477 = 0.301 + 1.431 = 1.732$.
8. Calculate: $10 \times \log_{10} 54 = 10 \times 1.732 = 17.32$.
9. Find the floor: $\lfloor 17.32 \rfloor = 17$.
10. Add 1: $17 + 1 = 18$.

Revision Table: Logarithm Properties

Additional Information: Characteristic and Mantissa

When we calculate \( \log_{10} N \), the result is often a decimal number. This decimal number has two parts:

• The integer part is called the characteristic. For \( \log_{10} (54)^{10} = 17.32 \), the characteristic is 17.
• The decimal part is called the mantissa. For \( \log_{10} (54)^{10} = 17.32 \), the mantissa is 0.32.

For a positive number \(N > 1\), the number of digits in \(N\) is always equal to the characteristic of \( \log_{10} N \) plus 1. This is exactly the formula \( \lfloor \log_{10} N \rfloor + 1 \) that we used.

This method is very useful for finding the number of digits in very large numbers that are expressed in exponential form.