The position of significant digits and Logarithms relationship..... I am unable to solve the following question as i don't understand what the relationship is between significant figures and Logarithms. Q-If log_(10)(7)= 0.8451 then the position of the first significant figure of 7^(-20)

figoveck38

figoveck38

Answered question

2022-11-06

The position of significant digits and Logarithms relationship.....
I am unable to solve the following question as i don't understand what the relationship is between significant figures and Logarithms.
Q-If log 10 ( 7 ) = 0.8451 then the position of the first significant figure of 7 20 .
The answer is the position of the first significant figure is 17th.
My book solves it in the following method-
log 10 ( x ) = 20 log 10 ( 7 )
= 16.9020 = 17 + 1 0.9020 = 1 ¯ 7 .0980
so the position is 17.
I fail to understand how and why this has happened please explain this solution to me and the relationship between significant figures and Logarithms...

Answer & Explanation

embutiridsl

embutiridsl

Beginner2022-11-07Added 26 answers

Take a number like 0.00234 and rewrite it in scientific notation: 2.34 × 10 3 . Now apply log to it and use your log rules:
log ( 2.34 × 10 3 ) = log ( 2.34 ) + log ( 10 3 ) = log ( 2.34 ) 3
Now, notice that log ( 2.34 ) = 0.36921585... is between 0 and 1, and so log ( 0.00234 ) is between 3 and 2, and the lower number just happens to be the spot that the first significant digit is sitting.
This isn't a coincidence. Whenever you write a number in scientific notation: a × 10 n and apply log to that number, you get
log ( a ) + n .
Since 1 a < 10 (this is what it means to write something in scientific notation) I know that n log ( a ) + n < n + 1. So I can get the power n from the scientific notation of a number by applying log and taking the integer to the left.
On the other hand, the power n is also the spot where the first significant digit is sitting.

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