The position of significant digits and Logarithms relationship.....

Question

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-

$$\log_{10}(x)=-20\log_{10}(7)$$

$$=-16.9020 =-17+1-0.9020 = \bar17 .0980$$

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 1

Take a number like $0.00234$ and rewrite it in scientific notation: $2.34 \times 10^{-3}$. Now apply $\log$ to it and use your log rules:

$$\log(2.34 \times 10^{-3}) = \log(2.34) + \log(10^{-3}) = \log(2.34) - 3$$

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: $a \times 10^n$ and apply $\log$ to that number, you get $$\log(a) + n.$$ Since $1 \leq a < 10$ (this is what it means to write something in scientific notation) I know that $n \leq \log(a )+ n < n+1.$ 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.