Question 12
The magnitude of an earthquake can be modelled by the logarithmic equation $M_A=\log_{10}\left(\dfrac{I_A}{I_0}\right)$, where $M_A$ is the magnitude at a location A, $I_A$ is the intensity of the earthquake at location A and $I_0$ is a constant.
An earthquake at location P had a magnitude of 5.2.
A different earthquake at location Q had a magnitude of 3.5.
Determine an equation involving logarithms that expresses the difference in magnitudes between the earthquakes at locations P and Q.
[1 mark]How many times more intense was the earthquake at location P than the earthquake at location Q?
[4 marks]A gap of 1.7 is not 1.7 times
Watch the two magnitudes become one log equation. Then see why a gap on the magnitude scale turns into a power of 10.
Magnitudes are logs. Subtracting logs divides the intensities, so a gap in magnitude becomes a power of 10.
Turn the gap into a number of times
Each magnitude sits above its intensity, measured in multiples of $I_0$. Move the magnitudes and watch how far apart the intensities are. Then choose how to turn the gap into an answer.
Choose how you would turn the gap in magnitudes into a number of times. The intensity scale shows the true answer to compare with.
That is the log written in index form. With the question’s magnitudes, P was about 50.1 times as intense as Q, which matches the intensity scale.
Right method, and it matches the intensity scale for any two magnitudes. Go back to the question’s magnitudes to get its answer.
The gap is a difference of two logs, not a ratio of intensities. Compare your answer with the ratio read from the intensity scale.
Dividing the magnitudes compares the logs, not the intensities. Logs turn division into subtraction, so it is the gap between the magnitudes that matters.
Each step of 1 on the magnitude scale multiplies the intensity by 10. So the gap goes in the power of 10, not next to it.
Both give 1 here, but only because the two magnitudes are equal. Move either slider and dividing the magnitudes no longer matches the intensity scale. Logs turn division into subtraction, so use the gap.
Both give 10 here, but only because the gap is exactly 1 and $10\times1=10^1$. Move either slider and ten times the gap no longer matches the intensity scale. The gap goes in the power of 10.
Q is now the bigger earthquake, so the gap is negative and P is only a fraction as intense as Q.
Work through the solution one mark at a time
Try each step yourself before you reveal it. Everything is algebra until the last line, where your calculator evaluates a power of 10.
Write the model once for each location. The letter A is replaced by P, then by Q. $I_0$ is a constant, so it is the same in both.
$$M_P=\log_{10}\left(\frac{I_P}{I_0}\right)$$
$$M_Q=\log_{10}\left(\frac{I_Q}{I_0}\right)$$
“The difference in magnitudes” means subtract one from the other.
$$\begin{aligned}M_P-M_Q&=\log_{10}\left(\frac{I_P}{I_0}\right)\\&\quad-\log_{10}\left(\frac{I_Q}{I_0}\right)\end{aligned}$$
Equivalent equations are accepted, such as the same right side with $5.2-3.5$ or $1.7$ on the left. Every example the guide accepts keeps the logarithms, and the question asks for an equation involving logarithms, so don’t stop at $5.2-3.5=1.7$.
Marker: correctly determines an equation involving the difference in magnitudes
Start from your part a) equation. The quotient law, $\log_a(m)-\log_a(n)=\log_a\left(\frac{m}{n}\right)$, turns the two logs into one. Then the $I_0$ cancels.
$$\begin{aligned}M_P-M_Q&=\log_{10}\left(\frac{\;\frac{I_P}{I_0}\;}{\frac{I_Q}{I_0}}\right)\\M_P-M_Q&=\log_{10}\left(\frac{I_P}{I_Q}\right)\end{aligned}$$
Equivalent equations are accepted, such as $5.2-3.5=\log_{10}\left(\frac{I_P}{I_Q}\right)$. Follow-through applies if part a) had a slip. Either this mark or the next one can be implied by your later working.
Marker: determines a simplified logarithmic equation
“How many times more intense” is asking for the ratio $\frac{I_P}{I_Q}$. To get it out of the log, change to index form: $\log_{10}(x)=y$ means $x=10^y$.
$$\frac{I_P}{I_Q}=10^{M_P-M_Q}$$
Marker: states an equation with intensities made the subject, i.e. the log equation is converted to an index equation
Now substitute the two magnitudes from the question.
$$\begin{aligned}\frac{I_P}{I_Q}&=10^{5.2-3.5}\\&=10^{1.7}\end{aligned}$$
Equivalent forms are accepted, such as $I_P=10^{1.7}\times I_Q$.
Marker: substitutes the given magnitude values appropriately
Evaluate the power on your calculator. Type 10, press the power key, then type 1.7. Typing $10^{5.2-3.5}$ with brackets around the power gives the same value.
$$\begin{aligned}\frac{I_P}{I_Q}&=50.1187\ldots\\I_P&\approx50.1187\,I_Q\end{aligned}$$
Appropriate rounding of the value is accepted. Finish with a sentence that answers the question in words.
Marker: determines how more intense the earthquake is at location P compared with location Q
Each phrase of the question gave you something
You are given the model and both magnitudes. Writing the model twice, subtracting, using the quotient law and changing to index form are all routines you have practised. The calculator only evaluates a power of 10.
If I had to guess where you might lose marks, it would be the jump from 1.7 to the answer. A gap of 1.7 is not 1.7 times as intense, and it is not 17 times. Each 1 on the magnitude scale is a factor of 10, so the answer is $10^{1.7}\approx50.1$. The other risk is part a). It asks for an equation involving logarithms, so write both logs out in full rather than only $5.2-3.5=1.7$.
Only mark this done when you could do it without help. Reading the solution does not count.
Question wording and marking-guide steps are from the 2024 QCAA Mathematical Methods external assessment, © State of Queensland (QCAA) 2024, licensed under CC BY 4.0, and have been adapted. Tangent Tuition is not affiliated with the QCAA.
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