Two sounds are measured at levels that differ by ten decibels. An earthquake is reported as one magnitude unit higher than another. In both situations, the numbers look as though they are separated by only a small amount.
But the underlying physical ratios can be much larger.
This is one reason to study Additional Mathematics: logarithms teach students how to interpret scales where equal numerical steps represent multiplication rather than ordinary addition.
The same mathematical idea appears in acoustics, earthquake measurements, scientific signals and other quantities spanning large ranges. It explains how modest-looking differences on a scale can represent ratios of ten, one hundred or more.
For Singapore Secondary 3 and Secondary 4 students, exponential and logarithmic functions are not merely an exercise in changing bases on a calculator. They are a way of understanding what a scale measures, how ratios behave and why a numerical label should not be interpreted as an ordinary percentage.
Start with an ordinary linear scale
Imagine one object has a mass of 5 kilograms and another has a mass of 10 kilograms.
Their difference is:
10 – 5 = 5 kilograms.
Their ratio is:
10/5 = 2.
The second object has twice the mass of the first.
If a third object has a mass of 15 kilograms, the increase from 10 to 15 is another 5 kilograms.
The absolute differences are equal, but the ratios are different:
10/5 = 2,
while:
15/10 = 1.5.
On a linear scale, equal numerical increments correspond to equal absolute changes in the represented quantity.
Logarithmic scales are constructed differently.
They are often chosen when ratios matter more than raw differences.
Why scientists use powers of ten
Consider the sequence:
1, 10, 100, 1000, 10000.
Each number is ten times the previous number.
Their common logarithms, using base ten, are:
log₁₀1 = 0.
log₁₀10 = 1.
log₁₀100 = 2.
log₁₀1000 = 3.
log₁₀10000 = 4.
A multiplication by ten becomes an addition of one on the logarithmic scale.
That is why logarithms are useful for describing quantities that span many orders of magnitude.
The original values range from one to ten thousand.
Their logarithms range only from zero to four.
The transformation compresses a very broad multiplicative range into a smaller numerical range.
The compression does not remove the differences. It encodes them in another form.
The fundamental logarithm identity
For a positive ratio r:
log₁₀(10r) = log₁₀10 + log₁₀r.
Since log₁₀10 = 1:
log₁₀(10r) = 1 + log₁₀r.
Thus multiplying the original quantity by ten adds one to its logarithm.
More generally, for positive a and b:
log₁₀(ab) = log₁₀a + log₁₀b.
And:
log₁₀(a/b) = log₁₀a – log₁₀b.
These are not unrelated rules to be memorised blindly.
They explain how a logarithmic scale converts multiplicative comparisons into additive differences.
A-Math gives students practice applying these identities while respecting positive-argument conditions.
What a decibel comparison represents
For a sound-intensity level comparison, a standard mathematical relationship is:
ΔL = 10 log₁₀(I₂/I₁),
where I₁ and I₂ are positive sound intensities measured consistently and ΔL is their level difference in decibels.
The ratio I₂/I₁ is dimensionless.
That matters because we take the logarithm of a ratio, not a bare physical unit.
If I₂ = 10I₁, then:
ΔL = 10 log₁₀10.
Therefore:
ΔL = 10 decibels.
So an increase of ten decibels in this intensity-level comparison corresponds to ten times the intensity.
The United States Centers for Disease Control and Prevention explains this logarithmic property in its NIOSH introduction to noise and hearing loss.
The mathematical relationship is about intensity, not a guarantee that humans experience the new sound as ten times as loud.
Twenty decibels represent a different factor
Suppose a sound intensity rises from I₁ to I₂ and the corresponding intensity-level difference is twenty decibels.
Use:
20 = 10 log₁₀(I₂/I₁).
Divide by ten:
2 = log₁₀(I₂/I₁).
Convert to exponential form:
I₂/I₁ = 10².
Therefore:
I₂/I₁ = 100.
A twenty-decibel increase corresponds to a hundredfold increase in intensity under the stated definition.
It is not merely twice the intensity increase associated with ten decibels.
The decibel increments add, but the underlying intensity ratios multiply.
This is exactly the kind of mathematical relationship students learn through logarithm laws.
Thirty decibels and powers of ten
Now suppose:
ΔL = 30 decibels.
Then:
30 = 10 log₁₀(I₂/I₁).
So:
log₁₀(I₂/I₁) = 3.
Therefore:
I₂/I₁ = 10³ = 1000.
A thirty-decibel difference corresponds to a factor of one thousand in intensity.
The pattern is clear:
A ten-decibel difference corresponds to a factor of ten.
Twenty decibels corresponds to a factor of one hundred.
Thirty decibels corresponds to a factor of one thousand.
Each additional ten-decibel step multiplies the intensity ratio by another ten.
This is a mathematical reason scientists use logarithmic scales to represent large ranges.
The values remain manageable even when the physical ratios become very large.
Doubling intensity produces about three decibels
Suppose the sound intensity doubles:
I₂/I₁ = 2.
Then:
ΔL = 10 log₁₀2.
Since log₁₀2 is approximately 0.30103:
ΔL ≈ 3.0103 decibels.
So doubling sound intensity corresponds to an increase of approximately three decibels in this intensity-level calculation.
This is not an exact equality of three decibels.
The exact expression is 10 log₁₀2.
A student should learn to distinguish a convenient approximate rule from the exact logarithmic relationship.
That distinction matters when several numerical transformations are combined.
For careful calculations, retain the logarithmic expression until a decimal answer is required.
What happens if intensity is halved?
Now let:
I₂/I₁ = 1/2.
Use the same relationship:
ΔL = 10 log₁₀(1/2).
Since:
log₁₀(1/2) = -log₁₀2,
we obtain:
ΔL = -10 log₁₀2.
Therefore:
ΔL ≈ -3.0103 decibels.
A negative level difference indicates a decrease relative to the initial intensity.
It does not mean sound intensity itself has become negative.
Both I₁ and I₂ remain positive.
The logarithm of a ratio below one is negative.
This is a useful illustration of negative logarithms: they can represent positive quantities that are smaller than a chosen reference.
Sound intensity and perceived loudness are not identical
A-Math can calculate sound-intensity ratios accurately.
Human perception is another subject.
A change of ten decibels corresponds to ten times the physical intensity in the stated comparison. That does not mean everyone hears the sound as precisely ten times as loud.
Perceived loudness is influenced by frequency, duration, hearing characteristics and other conditions.
Similarly, a physical intensity comparison is not by itself a complete hearing-safety assessment.
Relevant workplace noise standards consider how sound is measured and the exposure conditions.
Students should therefore keep the mathematical ratio separate from claims about subjective experience or safe exposure.
The CDC/NIOSH information on occupational noise provides additional context.
The central mathematical skill is interpreting what the logarithmic variable actually represents.
Decibels are not all one identical measurement
A decibel expresses a logarithmic ratio.
The quantity used in the ratio and the reference convention matter.
For sound intensity or power-like quantities, the formula involves a factor of ten before the logarithm.
For sound pressure level under the usual acoustic convention, the relationship can use a factor of twenty times the logarithm of a positive pressure-amplitude ratio.
This reflects the squared relationship between pressure amplitude and intensity under suitable physical conditions.
A student should not blindly replace sound pressure with intensity and assume the same formula without examining the definitions.
The mathematics is closely related, but the physical quantities are different.
This is an enrichment distinction that helps prevent misuse of an otherwise correct index law.
A pressure-ratio example
Suppose the relevant sound pressure amplitude doubles while other conditions allow the usual pressure-level relationship:
ΔL_p = 20 log₁₀(p₂/p₁).
Set:
p₂/p₁ = 2.
Then:
ΔL_p = 20 log₁₀2.
Therefore:
ΔL_p ≈ 6.0206 decibels.
That differs from the approximately three-decibel change for doubling intensity.
The reason is not that logarithms have changed.
The physical relationship uses a squared quantity, leading to the factor twenty for pressure amplitude.
This demonstrates why reading the variable definition is essential before substituting.
Students should not memorise one decibel shortcut and apply it to every physical measure.
A logarithmic scale can also describe earthquakes
Earthquake magnitude is another familiar example of logarithmic reasoning.
The United States Geological Survey explains that a one-unit magnitude increase on amplitude-based magnitude scales corresponds to a factor of ten in the measured amplitude used in that scale.
The associated energy comparison is much larger: approximately 32 times the energy release for a one-unit magnitude increase under the commonly used approximate energy relationship.
These are different ratios.
A magnitude number is not simply a count of how strongly someone felt shaking at one location.
Modern earthquake analysis also uses moment magnitude and other physical measurements rather than treating all scales as one interchangeable rule.
The USGS explanation of magnitude, energy and shaking intensity is a reliable starting point.
A one-unit magnitude comparison
Consider two earthquakes with magnitudes M₁ = 4 and M₂ = 5 under a compatible logarithmic amplitude-comparison convention.
Their difference is:
ΔM = 5 – 4 = 1.
The simplified amplitude ratio is:
10^(ΔM).
Therefore:
10¹ = 10.
The measured amplitude ratio associated with that unit change is ten.
It would be incorrect to say the second is only 25 percent larger because five is 25 percent larger than four numerically.
The magnitude labels are logarithmic.
A difference in labels is not a proportional difference in the underlying measurement.
This example illustrates why the representation of a quantity matters.
Two magnitude units mean a hundredfold amplitude factor
Now compare magnitude 4 with magnitude 6.
The difference is:
ΔM = 6 – 4 = 2.
Under the same amplitude-comparison convention:
Amplitude ratio = 10² = 100.
A magnitude-six event therefore corresponds to a hundredfold amplitude ratio relative to magnitude four in that idealised scale comparison.
The numerical magnitude increased by two.
The corresponding amplitude comparison multiplied by one hundred.
The exponential relationship explains that difference.
But it does not say that shaking at every location must be one hundred times as strong.
Observed shaking depends on factors such as distance, depth and local ground conditions.
Magnitude and local shaking intensity should be kept distinct.
Energy changes by a different power
USGS describes an approximate relationship under which a one-unit increase in earthquake magnitude corresponds to about 32 times the energy released.
Mathematically, the more precise common approximation is:
Energy ratio ≈ 10^[1.5(M₂ – M₁)].
For a difference of one magnitude unit:
Energy ratio ≈ 10^1.5.
This is approximately 31.62.
For a difference of two units:
Energy ratio ≈ 10³ = 1000.
So a two-unit increase corresponds to about a thousandfold energy ratio under this approximation.
That is not the same as the hundredfold amplitude comparison.
Students must not interchange amplitude and energy just because both are associated with earthquake magnitudes.
They measure different physical aspects.
Why earthquake energy is not simply amplitude squared here
Students may be tempted to reason that a tenfold amplitude must always mean one hundred times the released energy.
That would be an unjustified simplification across earthquake sources and measurements.
The approximate earthquake magnitude-energy relationship is based on seismological conventions and empirical relationships involving source properties.
Its common factor of approximately 31.6 per magnitude unit follows the exponent 1.5, not a universal square law for every seismological quantity.
To learn this responsibly, use the relationship explicitly supplied by a reputable source rather than invent a shortcut from an analogy with a different physical phenomenon.
The point of the A-Math lesson is to calculate with the stated model and preserve the meaning of its variables.
Compare magnitude values differing by 1.5
Let:
M₁ = 5,
and:
M₂ = 6.5.
Then:
ΔM = 1.5.
The simplified amplitude factor is:
10^1.5 = 10√10.
This is approximately 31.62.
For the approximate energy ratio:
10^[1.5(1.5)] = 10^2.25.
This is approximately 177.83.
The same magnitude difference produces different multiplicative comparisons for amplitude and energy.
Students should state which ratio is being calculated and which empirical approximation is used.
A bare answer such as “about 32 times bigger” would be ambiguous without specifying the physical measure.
Clear mathematical communication prevents that ambiguity.
The meaning of an earthquake’s local intensity
Earthquake magnitude describes an overall property of an earthquake source under the applicable scale.
The intensity of shaking experienced at a particular place can vary with location and conditions.
Two people at different distances from the same earthquake may experience different shaking intensities.
A mathematical model of magnitude does not directly give an exact local damage level or personal safety prediction.
The USGS guidance makes the distinction between magnitude, energy release and shaking intensity.
For students, this is a lesson in quantitative language.
Before comparing numbers, identify what the numbers actually measure.
Additional Mathematics supplies useful logarithm tools, not a complete model of earthquake hazards.
The inverse problem: recover a scale difference from a ratio
Suppose two sound intensities have ratio:
I₂/I₁ = 50.
The level difference is:
ΔL = 10 log₁₀50.
We can write:
50 = 5 × 10.
Therefore:
log₁₀50 = log₁₀5 + 1.
Numerically:
log₁₀50 ≈ 1.69897.
So:
ΔL ≈ 16.99 decibels.
The intensity ratio fifty is between ten and one hundred.
Therefore the level difference should lie between ten and twenty decibels.
Our result satisfies that bound.
This is an example of using order-of-magnitude reasoning to check a logarithmic calculation.
Recover the physical ratio from decibels
Now reverse the problem.
Suppose an intensity-level difference is 17 decibels.
Use:
17 = 10 log₁₀r,
where r is the positive intensity ratio.
Divide by ten:
1.7 = log₁₀r.
Convert to exponential form:
r = 10^1.7.
Numerically:
r ≈ 50.12.
Thus a seventeen-decibel difference corresponds to approximately fifty times the intensity under the defined comparison.
This is a direct link between logarithms and exponentials.
One function compresses a ratio into a level difference.
The inverse operation recovers the ratio.
Students who understand both directions can solve more than a one-template question.
Why two logarithmic differences can be added
Suppose intensity B is ten times intensity A.
Suppose intensity C is one hundred times intensity B.
Then:
I_B/I_A = 10.
I_C/I_B = 100.
The combined ratio is:
I_C/I_A = (I_C/I_B)(I_B/I_A).
Therefore:
I_C/I_A = 100 × 10 = 1000.
On the intensity-level scale, the first difference is ten decibels.
The second is twenty decibels.
Their sum is thirty decibels.
This is consistent with the thousandfold combined ratio:
10 log₁₀1000 = 30.
Multiplication of positive ratios becomes addition of logarithmic level differences.
That is the structural reason the arithmetic works.
Why directly adding two sound levels is different
Suppose two independent idealised sound sources each have the same intensity I at a chosen observation point, and their intensities combine additively under the assumptions of the example.
Their combined intensity is:
I_total = I + I = 2I.
The resulting intensity-level increase above either individual source is:
10 log₁₀2 ≈ 3.01 decibels.
So two equal-intensity contributions produce approximately a three-decibel increase in this simplified situation, not a doubling of the decibel number.
Adding sound levels numerically as though they were ordinary linear quantities would be incorrect.
The mathematical operation is to combine the underlying intensity contributions and then apply the logarithmic scale.
Real acoustic measurements may require additional considerations about coherence and measurement conventions.
Why a negative decibel difference can be valid
Suppose the new intensity is one-hundredth of the original intensity.
Then:
r = 1/100 = 10⁻².
The level difference is:
ΔL = 10 log₁₀(10⁻²).
Therefore:
ΔL = 10(-2) = -20 decibels.
The negative result does not indicate negative intensity.
It indicates that the new intensity is lower than the reference level.
This is similar to many relative logarithmic scales where the numerical level may be below zero while the underlying measured quantity remains positive.
The correct interpretation depends on the reference value and the quantity being compared.
A-Math students should distinguish a negative logarithm from a negative input to a logarithm.
The former can be valid; the latter is outside the real logarithm’s domain.
Logarithm input restrictions remain essential
The real logarithm log₁₀x is defined for x > 0.
In a sound-intensity comparison, the ratio I₂/I₁ must be positive.
For ordinary positive intensities, that condition is satisfied.
But if a student substitutes a negative intensity ratio into the real logarithm, the model is invalid.
Similarly, log₁₀0 is undefined as a finite real number.
As its positive input approaches zero, the logarithm decreases without bound.
A student should not interpret that as a defined logarithmic value at exactly zero.
These domain ideas are part of the function’s mathematical meaning.
A-Math teaches students to preserve them through transformations and calculator steps.
A common mistake in percentage comparisons
Someone says:
“Earthquake magnitude six is fifty percent greater than magnitude four, because six is one-and-a-half times four.”
That calculates a ratio between the printed labels.
It does not calculate the corresponding physical amplitude or energy ratio.
For a two-unit magnitude difference, the simplified amplitude factor is 100.
The approximate energy factor is 1000.
Those comparisons are based on differences on the logarithmic scale.
It is generally not meaningful to interpret the percentage difference between two magnitude labels as the percentage change in released energy.
Students should ask what the scale is measuring before using an ordinary percent formula.
The same care applies to decibel levels.
Scientific graphs sometimes use logarithmic axes
A logarithmic axis spaces values according to their logarithms rather than their raw magnitudes.
For example, positions labelled 1, 10, 100 and 1000 may be equally spaced along a base-ten logarithmic axis.
Those equal gaps represent equal multiplicative factors of ten.
A linear axis would instead place the values proportionally to their ordinary numerical differences.
Neither representation changes the underlying data.
But the visual appearance of a graph can change dramatically.
A student should inspect whether an axis is linear or logarithmic before interpreting the slope, shape or spacing of plotted measurements.
Otherwise a graph may appear to show steady growth when the original quantity is multiplying rapidly.
Exponential functions become straight lines on a logarithmic scale
Consider an idealised model:
y = 5 × 2^x,
where y is positive.
Take base-ten logarithms:
log₁₀y = log₁₀5 + x log₁₀2.
This has the form of a straight-line equation in x and log₁₀y.
Its gradient is log₁₀2.
Its vertical intercept is log₁₀5.
Thus a pure exponential relationship can become a straight line when the output is represented logarithmically.
This is a useful connection between exponential functions and straight-line graphs.
However, an apparently straight transformed graph does not automatically validate a real-world exponential model.
The quality, range and uncertainties of the measurements still matter.
A note about earthquake and sound references
A scientific statement should cite the definition of its scale, not rely on a classroom slogan alone.
For sound, a source should identify the level quantity and appropriate intensity, power or pressure convention.
For earthquakes, a source should distinguish the magnitude scale from local shaking intensity and the approximate energy relationship.
Two reliable starting points are the CDC/NIOSH explanation of decibels and the USGS overview of earthquake magnitude and energy.
They support the numerical comparisons used as illustrations.
The mathematics teaches how to calculate from the stated definitions. Subject specialists determine which definitions and measurements apply to a particular scientific task.
An effective four-step calculation check
First, identify the physical ratio or level difference.
Second, choose the correct logarithmic relationship for the quantity.
Third, perform the algebra, keeping the logarithm’s positive-input condition.
Fourth, interpret the result using the original units and physical meaning.
For example, with a sound intensity ratio of 100:
ΔL = 10 log₁₀100 = 20 decibels.
The ratio is dimensionless, while the result is a level difference.
The statement is about a hundredfold intensity ratio, not a hundredfold perceived loudness.
Writing that final interpretation prevents the numerical answer from being given a meaning the equation does not support.
Four common errors and how to repair them
Mistake one: treating logarithmic labels as linear quantities
A student says a two-unit increase in magnitude represents an addition of two physical units.
The repair is to use the scale’s exponential relationship and calculate the corresponding ratio.
Mistake two: confusing physical measures
The learner reports the earthquake amplitude factor as though it were the energy factor.
The repair is to label which quantity is being compared and use its appropriate relationship.
Mistake three: applying the wrong decibel coefficient
The student uses a pressure-amplitude expression to calculate an intensity ratio without checking the definition.
The repair is to define the measured physical quantity before selecting the formula.
Mistake four: turning a level difference into perceived loudness
The student assumes a ten-decibel increase always sounds precisely ten times as loud.
The repair is to distinguish a mathematical intensity ratio from a subjective perception measure.
These errors look numerical, but their deeper cause is confusion about what the variables represent.
A five-session learning plan
Session one: master common logarithms
Calculate log₁₀ of 1, 10, 100, 1000 and 10000.
Explain why equal logarithmic increments correspond to equal multiplication factors.
Session two: use sound intensity ratios
Calculate level differences for intensity factors 2, 10, 50 and 100.
Then reverse the problem by recovering a ratio from a stated decibel difference.
Session three: compare earthquake magnitudes
Use differences of one, two and 1.5 magnitude units.
Calculate the simplified amplitude factors and approximate energy factors separately.
State clearly which relationship is being used.
Session four: inspect graph axes
Compare the same exponential data plotted on ordinary and logarithmic output scales.
Explain why the transformed relationship can appear straight.
Session five: solve unfamiliar mixed questions
Mix sound-level ratios, logarithmic equations and exponential graph transformations.
Before calculating, identify the base, reference quantity, units and permitted domain.
The aim is transfer, not memorising isolated numbers such as ten or thirty-two.
What parents can ask during revision
Ask what a logarithmic scale does to multiplication.
Ask why a difference of twenty decibels does not mean twenty times the intensity.
Ask why an earthquake magnitude difference should not be interpreted as an ordinary percentage difference between its printed labels.
Ask which quantity is measured by the formula: intensity, pressure, amplitude or energy.
Ask what happens when the physical ratio falls below one.
These questions encourage mathematical interpretation.
A student who can explain the ratio behind a scale label has understood more than how to operate a logarithm key.
Singapore’s G3 Additional Mathematics connection
The 2027 Singapore-Cambridge Secondary Education Certificate G3 Additional Mathematics syllabus, subject K341, includes exponential and logarithmic functions, logarithm laws and their use in modelling relationships.
The sound and earthquake examples illustrate applications of those foundations. Detailed acoustics and seismology are enrichment rather than separate G3 A-Math syllabus topics.
Students and parents can consult the official SEAB 2027 G3 syllabus directory to check the current assessed scope.
The teaching goal is to understand how logarithmic functions encode physical ratios, not to make safety decisions from simplified calculations.
How a three-student tutorial can help
One learner may use logarithm laws correctly but confuse a level difference with an intensity ratio.
Another may know the exponential conversion but omit the factor of ten in a decibel formula.
A third may calculate the earthquake amplitude factor correctly while confusing it with energy release.
These are different difficulties.
In a carefully managed three-student Additional Mathematics tutorial, a tutor can ask the learner to identify the measured quantity before applying a formula.
The immutable Secondary 1 Mathematics small-group tutorial in Clementi illustrates why number sense, ratios and clear algebraic explanations are important foundations.
Students progressing effectively through school and independent practice may not require additional tuition. Support is most useful when an identifiable misconception keeps returning.
Frequently asked questions
What does a ten-decibel sound-intensity difference mean?
Under the standard intensity-level comparison, it corresponds to a tenfold intensity ratio.
Does twenty decibels represent twenty times the intensity?
No. A twenty-decibel difference corresponds to a factor of one hundred in the standard intensity-ratio calculation.
Why does doubling sound intensity give about three decibels?
Because 10 log₁₀2 is approximately 3.0103.
Is earthquake magnitude the same as shaking intensity at my location?
No. Magnitude describes an earthquake source under its scale, while local shaking intensity depends on the location and conditions.
Why do earthquake amplitude and energy ratios differ?
They describe different physical quantities and use different relationships. A one-unit magnitude difference is associated with a tenfold amplitude factor on compatible amplitude scales and roughly 31.6 times energy under the common approximation.
Can a logarithm have a negative value?
Yes. For example, log₁₀0.1 = -1, even though the input 0.1 is positive.
Is a logarithmic scale required knowledge in Additional Mathematics?
Exponential and logarithmic laws are part of the subject. These particular scientific applications are explanatory enrichment.
The lasting reason to understand logarithmic scales
Additional Mathematics teaches students that a small difference between two labels does not always represent a small difference in the underlying quantity.
A logarithm turns multiplication into addition.
Its inverse converts a level difference back into a ratio.
Those relationships are useful for interpreting scientific measurements spanning enormous ranges.
The more important habit is to ask what the scale measures and what its numbers mean.
That makes the mathematics both more accurate and more useful.
For connected reading, explore Why Study Additional Mathematics | Use Logarithms to Find How Long Growth Takes, Why Study Additional Mathematics | Understand Half-Life Through Exponential Decay, Why Study Additional Mathematics | Estimate Whether an Answer Is Reasonable and the eduKateSG Additional Mathematics project library.