The intensities of earthquakes are measured with seismographs all over the world at different distances from the epicenter. Suppose that the intensity of a medium earthquake is originally reported as $$ 10^{5.5} $$ times $$ I_{0} $$. Later this value is revised as $$ 10^{5.6} $$ times $$ I_{0} $$. The local magnitude $$ M $$ (on the Richter scale) of an earthquake of intensity $$ I $$ is given by $$ \log \left(\frac{I}{I_{0}}\right) $$. Part: $$ 0 / 3 $$ Part 1 of 3 (a) Determine the magnitude of the earthquake using the original estimate for intensity. $$ M=\square $$

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The magnitude of the earthquake using the original estimate for intensity is 5.5. To determine the magnitude of the earthquake using the original estimate for intensity, we can use the formula for local magnitude $$M$$: $$M = \log\left(\frac{I}{I_0}\right)$$ Given that the intensity of the earthquake is originally reported as $$10^{5.5}$$ times $$I_0$$, we can substitute this value into the formula: $$M = \log\left(\frac{10^{5.5} \cdot I_0}{I_0}\right)$$ Simplifying the expression inside the logarithm: $$M = \log\left(10^{5.5}\right)$$ Using the property of logarithms that $$\log(a^b) = b \cdot \log(a)$$: $$M = 5.5 \cdot \log(10)$$ Since $$\log(10) = 1$$, we can simplify further: $$M = 5.5 \cdot 1$$ Therefore, the magnitude of the earthquake using the original estimate for intensity is: $$M = 5.5$$

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