Question

benfords law states that the probability that a number in a set has a given leading digit, d, is ( p(d)=log (d + 1)-log (d) ). state which property you would use to rewrite the expression as a single logarithm, and rewrite the logarithm. what is the probability that the number 1 is the leading digit? explain.

Explanation:

Step1: Apply logarithm property

Use the quotient rule of logarithms: $\log a-\log b=\log\frac{a}{b}$.
So, $P(d)=\log(d + 1)-\log(d)=\log\frac{d + 1}{d}$.

Step2: Calculate probability for \(d = 1\)

Substitute \(d = 1\) into \(P(d)=\log\frac{d + 1}{d}\).
We get \(P(1)=\log\frac{1+1}{1}=\log2\).
Using a calculator, \(\log2\approx0.3010\).

Answer:

The quotient rule of logarithms is used. The single - logarithm form is \(P(d)=\log\frac{d + 1}{d}\). The probability that \(1\) is the leading digit is \(P(1)=\log2\approx0.3010\) because when \(d = 1\), substituting into the formula \(P(d)=\log\frac{d + 1}{d}\) gives \(P(1)=\log\frac{2}{1}=\log2\).