by what factor does the intensity increase for each whole - number increase in the richter scale? use your…

Explanation:

Step1: Recall Richter - scale formula

The Richter - scale formula is $M=\log\left(\frac{I}{I_0}\right)$, where $M$ is the magnitude on the Richter scale, $I$ is the intensity of the earthquake, and $I_0$ is a reference intensity.

Step2: Let $M_1=\log\left(\frac{I_1}{I_0}\right)$ and $M_2=\log\left(\frac{I_2}{I_0}\right)$

Suppose $M_2 = M_1+1$. Then $\log\left(\frac{I_2}{I_0}\right)=\log\left(\frac{I_1}{I_0}\right)+1$.

Step3: Use logarithm properties

Since $\log a-\log b=\log\frac{a}{b}$ and $1 = \log10$, we have $\log\left(\frac{I_2}{I_0}\right)-\log\left(\frac{I_1}{I_0}\right)=\log10$. Then $\log\left(\frac{\frac{I_2}{I_0}}{\frac{I_1}{I_0}}\right)=\log10$.

Step4: Simplify the left - hand side

$\frac{\frac{I_2}{I_0}}{\frac{I_1}{I_0}}=\frac{I_2}{I_1}$, so $\log\left(\frac{I_2}{I_1}\right)=\log10$.

Step5: Use the one - to - one property of logarithms

If $\log a=\log b$, then $a = b$. So $\frac{I_2}{I_1}=10$.

Answer:

The intensity increases by a factor of 10 for each whole - number increase in the Richter scale.