How many digits are there in (54) 10 ? (Given that log 10 2 = 0.301 and log 10 3 = 0.477)

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 Identify the number $N = (54)^{10}$. Recall the formula for the number of digits: $\lfloor \log_{10} N \rfloor + 1$. Calculate $\log_{10} (54)^{10}$ using logarithm properties. $\log_{10} (54)^{10} = 10 \times \log_{10} 54$. Break down $\log_{10} 54$: $\log_{10} 54 = \log_{10} (2 \times 3^3) = \log_{10} 2 + 3 \log_{10} 3$. Substitute given values: $\log_{10} 2 = 0.301$, $\log_{10} 3 = 0.477$. Calculate: $\log_{10} 54 = 0.301 + 3 \times 0.477 = 0.301 + 1.431 = 1.732$. Calculate: $10 \times \log_{10} 54 = 10 \times 1.732 = 17.32$. Find the floor: $\lfloor 17.32 \rfloor = 17$. Add 1: $17 + 1 = 18$. Revision Table: Logarithm Properties Property Formula Product Rule $ \log_b (MN) = \log_b M + \log_b N $ Power Rule $ \log_b (M^p) = p \log_b M $ Change of Base $ \log_b M = \frac{\log_c M}{\log_c b} $ Logarithm of Base $ \log_b b = 1 $ Logarithm of 1 $ \log_b 1 = 0 $ 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.

Find Number of Digits (54)^10 Using Logarithms

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CDS - 2020

Mathematics

How many digits are there in (54) ¹⁰? (Given that log ₁₀2 = 0.301 and log ₁₀3 = 0.477)

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CDS I 2020 Elementary Mathematics Previous Year Paper (02-Feb-2020)

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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

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

Revision Table: Logarithm Properties

Property	Formula
Product Rule	$ \log_b (MN) = \log_b M + \log_b N $
Power Rule	$ \log_b (M^p) = p \log_b M $
Change of Base	$ \log_b M = \frac{\log_c M}{\log_c b} $
Logarithm of Base	$ \log_b b = 1 $
Logarithm of 1	$ \log_b 1 = 0 $

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.

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