Number of digits in Exponentiation

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

#NumberTheory #HelpMe! #MathProblem #Math

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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

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 - 7 years, 6 months ago

In term of a formula, the number of digits of $n$ is equal to $\lfloor \log_{10} n \rfloor + 1$ .

Mike Kong - 7 years, 6 months ago

How can we do it without using Log Tables?

Akshat Jain - 7 years, 6 months ago

Ya! Do U know?

Swapnil Das - 6 years, 2 months ago

Thanks

Mayank Kaushik - 7 years, 6 months ago

Use logarithm.

Aditya Parson - 7 years, 6 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$