# Magnitude comparisons — optional authored practice

Subject: Disaster Management

Exam: TEE

Reader: https://deep-learning-study-room.vercel.app/dm/#tee/practice/calculations/magnitude

Canonical: https://deep-learning-study-room.vercel.app/read/dm/tee/practice/calculations--magnitude/

Supplementary worked examples · no numerical question found in the available 12 TEE sittings

**Source of the calculations.** M2 PDF p 55 teaches the tenfold amplitude relation. The numerical values below were added for explanation; none is a recovered PYQ. The approximate energy-ratio relation is an external scientific extension (USGS), not a formula explicitly found in the supplied lecture text.

**Definition.** A logarithmic scale expresses multiplicative changes as differences. A base-10 logarithm asks: “10 raised to what power gives this number?”

**Simple meaning.** Log bas ulta sawaal hai: 10 ki kaunsi power?

For example, 10² = 100, so log₁₀(100) = 2. On a compatible amplitude-based magnitude scale, a difference of 1 corresponds to a factor of 10 in measured amplitude after the required corrections. Earthquake energy follows the approximate relationship below; energy and amplitude are different quantities.



$$
\Delta M=M_2-M_1,\qquad \frac{A_2}{A_1}=10^{\Delta M}
$$





$$
\frac{E_2}{E_1}\approx10^{1.5\Delta M}
$$



A is wave amplitude; E is released energy; Δ means “change in”. Amplitude comparisons assume the relevant scale and comparable/corrected recording conditions. Do not turn an energy ratio into a claim about building damage.

### Illustrative calculation (not a PYQ). Compare the amplitude and approximate energy of earthquakes of magnitude 6 and 5.

#### Answer

## Given

M₂ = 6; M₁ = 5.

## To find

A₂/A₁ and E₂/E₁.

## Formula and why



$$
A_2/A_1=10^{\Delta M};\quad E_2/E_1\approx10^{1.5\Delta M}
$$



The magnitude difference is the exponent in the respective logarithmic relationship.

## Step-by-step solution

#### Step 1. Find the difference



$$
\Delta M=6-5=1
$$



#### Step 2. Find amplitude ratio



$$
A_2/A_1=10^1=10
$$



#### Step 3. Find energy ratio



$$
E_2/E_1\approx10^{1.5}=31.6228
$$



**Final answer:** 10 times the amplitude and approximately 31.6 times the energy.

#### Illustrative application · Same magnitude, different local effects

One earthquake has a source magnitude, but a nearby settlement with vulnerable buildings may experience stronger damaging effects than a distant settlement on firm ground. A one-unit magnitude difference means a tenfold amplitude ratio in the stated logarithmic comparison; it does not mean ten times the damage.

### Illustrative calculation (not a PYQ). Compare magnitude 7 with magnitude 5 using amplitude and energy ratios.

#### Answer

## Given

M₂ = 7; M₁ = 5.

## To find

The two ratios.

## Formula and why



$$
A_2/A_1=10^{\Delta M};\quad E_2/E_1\approx10^{1.5\Delta M}
$$



There are two magnitude units between the events, so multiply factors through the exponent.

## Step-by-step solution

#### Step 1. Subtract magnitudes



$$
\Delta M=7-5=2
$$



#### Step 2. Calculate amplitude



$$
10^2=100
$$



#### Step 3. Calculate energy



$$
10^{1.5\times2}=10^3=1000
$$



**Final answer:** 100 times the amplitude; approximately 1,000 times the energy.

#### Illustrative application · Same magnitude, different local effects

One earthquake has a source magnitude, but a nearby settlement with vulnerable buildings may experience stronger damaging effects than a distant settlement on firm ground. A one-unit magnitude difference means a tenfold amplitude ratio in the stated logarithmic comparison; it does not mean ten times the damage.

### Illustrative calculation (not a PYQ). A corrected amplitude ratio is 100. Find the magnitude difference. If the smaller event has magnitude 4.3, estimate the larger event’s magnitude.

#### Answer

## Given

A₂/A₁ = 100; M₁ = 4.3.

## To find

ΔM and M₂.

## Formula and why



$$
\Delta M=\log_{10}(A_2/A_1),\quad M_2=M_1+\Delta M
$$



The logarithm reverses exponentiation: 10² is 100.

## Step-by-step solution

#### Step 1. Take the base-10 logarithm



$$
\Delta M=\log_{10}(100)=2
$$



#### Step 2. Add the difference



$$
M_2=4.3+2=6.3
$$



**Final answer:** Magnitude difference 2; larger magnitude 6.3.

#### Illustrative application · Same magnitude, different local effects

One earthquake has a source magnitude, but a nearby settlement with vulnerable buildings may experience stronger damaging effects than a distant settlement on firm ground. A one-unit magnitude difference means a tenfold amplitude ratio in the stated logarithmic comparison; it does not mean ten times the damage.

**Calculator:** use the *log* key for base 10, not *ln*. For 101.5, use the calculator’s 10ˣ function with exponent 1.5. Keep brackets around a multiplied exponent.

**Yaad rakho:** +1 magnitude: ×10 amplitude, about ×32 energy.