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Number of digits in Exponentiation

How to find Number of digits in Exponentiation like number of digits in 6 ^ 200 or 74 ^100

Note by Mayank Kaushik
3 years, 9 months ago

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Since we write our numbers in base 10, we can determine the number of digits by finding our number as an equivalent power of 10. ie:

\(10^x=6^{200}\)

\(log_{10}(10^x)=log_{10}(6^{200})\)

\(x=200log_{10}(6)\)

\(x \approx 155.63\)

Since our answer is going to be a whole number (as it is an integer raised to a positive integer power), we can check a few examples and see that we need to round up to get the solution:

Number of digits of \(6^{200}=\boxed{156}\) Ryan Carson · 3 years, 9 months ago

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@Ryan Carson In term of a formula, the number of digits of \(n\) is equal to \(\lfloor \log_{10} n \rfloor + 1\). Mike Kong · 3 years, 9 months ago

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@Ryan Carson How can we do it without using Log Tables? Akshat Jain · 3 years, 9 months ago

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@Akshat Jain Ya! Do U know? Swapnil Das · 2 years, 4 months ago

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@Ryan Carson Thanks Mayank Kaushik · 3 years, 9 months ago

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Use logarithm. Aditya Parson · 3 years, 9 months ago

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