Multiple choice

Number of digits in the number $\displaystyle 5^{17}$ is ...............(Use approximate value of $\displaystyle \log _{10}2$ is 0.3010)

  1. $10$
  2. $11$
  3. $12$
  4. $13$
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C Correct answer
Explanation

Number of digits = floor(log10(5^17)) + 1. log10(5^17) = 17 * log10(5) = 17 * log10(10/2) = 17 * (1 - log10(2)) = 17 * (1 - 0.3010) = 17 * 0.699 = 11.883. floor(11.883) + 1 = 11 + 1 = 12.

AI explanation

To find the number of digits, we calculate the base-10 logarithm of the number. Using the logarithm power rule, log(5^17) equals 17 multiplied by log(5). We rewrite log(5) as log(10/2), which equals 1 minus 0.3010, giving 0.6990. Multiplying 17 by 0.6990 yields 11.883, and since the characteristic is 11, the number of digits is 11 plus 1, resulting in 12.

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