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Number Theory Level 3

What power of 8 has 1000 digits ?

Note : I have found this problem on internet but i dont understand its solution so i wanted an understandable solution

1 solution

Siddhartha Srivastava
Mar 14, 2014

Let the power of 8 be $8^x$ .

Note that taking $log_{10}$ both sides gives the number of digits minus one

Then $1000 > \log{8^x} \geq 999$

$\Rightarrow 1000 > x\log8 \geq 999$

$\Rightarrow 1000 > 3x\log2 \geq 999$

$\Rightarrow 333.3 > x \times 0.3010299 \geq 333$

$\Rightarrow 1107.309 > x > 1106.2026$

Therefore $x = 1107$ .

beautiful

K.v. Jupiter - 7 years, 2 months ago

thanks

Rishabh Jain - 7 years, 2 months ago