The Richter scale is used to measure the intensity of...

The Richter scale is used to measure the intensity of earthquakes. The Richter scale rating of an earthquake is given by the formula $$ R=\frac{2}{3}(\log E-11.8) $$ where $E$ is the energy released by the earthquake (measured in ergs ${ }^{34}$ ). a. The San Francisco earthquake of 1906 registered $R=8.2$ on the Richter scale. How many ergs of energy were released? b. In 1989 another San Francisco earthquake registered $7.1$ on the Richter scale. Compare the two: The energy released in the 1989 earthquake was what percentage of the energy released in the 1906 quake? c. Solve the equation given above for $E$ in terms of $R$. d. Use the result of part (c) to show that if two earthquakes registering $R_{1}$ and $R_{2}$ on the Richter scale release $E_{1}$ and $E_{2}$ ergs of energy, respectively, then $$ \frac{E_{2}}{E_{1}}=10^{1.5\left(R_{2}-R_{1}\right)} $$ e. Fill in the blank: If one earthquake registers 2 points more on the Richter scale than another, then it releases times the amount of energy.

The Richter scale is used to measure the intensity of earthquakes. The Richter scale rating of an earthquake is given by the formula
$$
R=\frac{2}{3}(\log E-11.8)
$$
where $E$ is the energy released by the earthquake (measured in ergs ${ }^{34}$ ).
a. The San Francisco earthquake of 1906 registered $R=8.2$ on the Richter scale. How many ergs of energy were released?
b. In 1989 another San Francisco earthquake registered $7.1$ on the Richter scale. Compare the two: The energy released in the 1989 earthquake was what percentage of the energy released in the 1906 quake?
c. Solve the equation given above for $E$ in terms of $R$.
d. Use the result of part (c) to show that if two earthquakes registering $R_{1}$ and $R_{2}$ on the Richter scale release $E_{1}$ and $E_{2}$ ergs of energy, respectively, then
$$
\frac{E_{2}}{E_{1}}=10^{1.5\left(R_{2}-R_{1}\right)}
$$
e. Fill in the blank: If one earthquake registers 2 points more on the Richter scale than another, then it releases times the amount of energy.

Key Concepts

Ratio and Proportional Reasoning

Ratio and proportional reasoning involve comparing quantities to understand their relative sizes. In logarithmic contexts, differences in scale readings translate into multiplicative differences in the original quantities. This concept is widely used in understanding how a fixed difference on the Richter scale corresponds to a specific multiplicative factor change in the energy released.

Properties of Logarithms

The properties of logarithms—such as the product, quotient, and power rules—allow for the manipulation and simplification of logarithmic expressions. These properties are essential for rearranging equations, solving for unknowns, and comparing logarithmically scaled quantities in various scientific and engineering contexts.

Exponential Equations

Exponential equations involve variables in the exponent and require applying logarithms to isolate the variable. Solving such equations is crucial when reversing logarithmic transformations, for example, converting a Richter scale reading back into the actual energy released by an earthquake.

Richter Scale

The Richter scale is a logarithmic scale used in seismology to quantify the amount of energy released by an earthquake. It converts the huge variations in seismic energy into manageable numbers by taking the logarithm of the energy, thereby allowing for a comparison of earthquake magnitudes in a compressed and intuitive format.

Logarithms

Logarithms are the inverse operations of exponentiation and are used to transform multiplicative relationships into additive ones. They help simplify calculations where quantities span several orders of magnitude, particularly in contexts like measuring seismic energy or other phenomena that exhibit exponential growth or decay.

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