Question

If log2=0.301,log3=0.477,log⁡2=0.301,log⁡3=0.477,find the number of digits in (108)10.

Answer 1

108*10
1080
4digit
this is the answer

Answer 2

Answer-

The number of digits in the given number is 21

Solution-

For any positive integer 'n', the number of digits in 'n' is,

$\left \lfloor \log_{10} n \right \rfloor+1$

Here,

$n=108^{10}$

Applying the formula, the number of digits in n will be,

$=\left \lfloor \log_{10} 108^{10}\right \rfloor+1$

$=\left \lfloor 10\times \log_{10} 108\right \rfloor+1$

$=\left \lfloor 10\log_{10} (2\times 2\times 3\times 3\times 3)\right \rfloor+1$

$=\left \lfloor 10(\log_{10} 2+\log_{10}  2+\log_{10} 3+\log_{10}3+\log_{10} 3)\right \rfloor+1$

$=\left \lfloor 10(0.301+0.301+0.477+0.477+0.477)\right \rfloor+1$

$=\left \lfloor 10(2.033)\right \rfloor+1$

$=\left \lfloor 20.33\right \rfloor+1$

$=20+1$

$=21$

Therefore, it has 21 digits.