Meh! Where's My Calculator?

P.S: Here , Every log's base is 10. Note that, $\log _{ 10 }{ 1\quad =\quad 0 }$ and, $\log _{ 10 }{ 10\quad =\quad 1 }$ also , $\log _{ 10 }{ 100\quad =\quad 2 }$

Have you noticed Something? when the number of digits were 1 the result of it's log was 0.

when the number of digits were 2 the result of it's log was 1. and

when the number of digits were 3 the result of it's log was 2.

have you noticed a pattern ? we can measure any number's number of digits using log !

now let us Create a Formula !

I want to find out how many digits are there in the number '1' we can easily tell it's 1 but we should find it using log!

for log 1=0 so, if we add one with it we can find the number of the digit! log 1 + 1 =1 ! now let's try it for 10 and 100.

log 10= 1 hence, log 10+1= 2 , So 10 has 2 digits.

log 100 =2 hence, log 100+1 = 3 , so 100 has 3 digits.

Now, What if the number is something like 2 or 5 or 126 or something?

log 2 = 0.301.. hence log 2+1=1.301... but it has 1 digit! so how can we make 1.301... into 1? we use "floor" ofcourse !! floor is a symbol like this $\left\lfloor x \right\rfloor$ It means the highest integer which is smaller than x. like $\left\lfloor 2.5 \right\rfloor =2\quad ,\quad \left\lfloor 2.11 \right\rfloor =2\quad and\quad \left\lfloor 2.9999 \right\rfloor =2$

so now we can say, $\left\lfloor \log _{ 10 }{ 2\quad +\quad 1 } \right\rfloor =1$

the same rule applies for all positive integer numbers ! So now we can simply say that , for any natural number 'N' , the number of digits N has is $\left\lfloor \log _{ 10 }{ N\quad +\quad 1 } \right\rfloor$

Now look at my solution once more ! If you still have problem understanding it let me know!

It's like this for anti-log there are two main parts.. The one before the decimals and the ones after the decimals.. Consider the log value is x.y (. represents decimal and not multiplication), after anti-log y gives the ans & (x+1) gives the number of digits after which the decimal is to be placed..

Have you checked my reply/response to Aditya Vikram's comment ? If you didn't please check ! After checking if u still have troubles understanding it tell me

Have you checked my reply/response to Aditya Vikram's comment ? If you didn't please check ! After checking if u still have troubles understanding it tell me