by what approximate factor is the intensity of an earthquake with magnitude 5.4 greater than an earthquake…

Explanation:

Step1: Set up equations for each magnitude

For magnitude $M_1 = 5.4$, we have $5.4=\log\left(\frac{I_1}{I_0}\right)$. Since $I_0 = 1$, then $5.4=\log(I_1)$ and $I_1 = 10^{5.4}$ by the definition of logarithm ($y=\log(x)$ implies $x = 10^{y}$). For magnitude $M_2=5.3$, we have $5.3=\log\left(\frac{I_2}{I_0}\right)$. Since $I_0 = 1$, then $5.3=\log(I_2)$ and $I_2 = 10^{5.3}$.

Step2: Find the ratio of intensities

We want to find the factor by which $I_1$ is greater than $I_2$, so we calculate the ratio $\frac{I_1}{I_2}$. Substituting $I_1 = 10^{5.4}$ and $I_2 = 10^{5.3}$ into the ratio, we get $\frac{10^{5.4}}{10^{5.3}}$. Using the exponent - rule $\frac{a^{m}}{a^{n}}=a^{m - n}$, we have $\frac{10^{5.4}}{10^{5.3}}=10^{5.4 - 5.3}=10^{0.1}$.

Step3: Calculate the value of $10^{0.1}$

$10^{0.1}\approx1.26$.

Answer:

C. 1.26