A quantity that loses half of what remains each hour never loses the same amount twice. Exponential models describe this shrinking pace. Logarithms let us run the question backward and infer how long a specified change takes.
These are tools for reasoning, not just calculator buttons. We will connect each tool to a physical question, derive its useful form and check what can go wrong when its assumptions are ignored.
What you will learn to do
- Distinguish constant loss from constant fractional loss.
- Interpret a time constant and solve an exponential for elapsed time.
- Use dimensionless logarithms and test transformed graphs.
- Convert angles to radians and estimate approximation errors.
- Retain the leading nonzero term in a small difference.
Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Exponential change, logarithms and controlled approximations Learning goals: - Distinguish constant loss from constant fractional loss. - Interpret a time constant and solve an exponential for elapsed time. - Use dimensionless logarithms and test transformed graphs. - Convert angles to radians and estimate approximation errors. - Retain the leading nonzero term in a small difference. Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
BEFORE YOU READ
Bring back one idea
Try a sentence or a sketch before opening the comparison. If this is new, use the explanation to get started.
How does a fixed fraction lost per interval differ from a fixed amount lost per interval?
Compare your reasoning
A fixed fraction acts on the remaining quantity, so equal intervals have equal ratios. A fixed amount gives equal differences. Those lead to exponential and linear models respectively.
Revisit the explanation →01 / Exponential change, logarithms and controlled approximations
Equal fractions, not equal amounts
A tank draining at a constant 2 litres per minute loses the same amount in every minute until its operating assumptions fail. A different process might lose a fixed fraction of what remains per minute. Then the loss becomes smaller as the remaining amount decreases. The second process is the starting idea behind an exponential, and confusing these two rules produces very different predictions.
Begin with discrete intervals. If a quantity retains three quarters of its value after each interval, it has fractions 1, 3/4, (3/4)² and (3/4)³ after zero, one, two and three intervals. Multiplication by the same factor replaces repeated subtraction of the same amount. The graph falls steeply at first and then flattens, without reaching zero in a finite number of these ideal steps.
A continuous exponential describes the limiting model in which the instantaneous rate is proportional to the current amount. This proportionality is a physical assumption, not a universal law of declining quantities. Constant leakage, changing environments or an additional source can invalidate a pure exponential model.
Differentiate the proposed solution to recover the rate equation. At t = 0 it returns the initial value y₀. The coefficient k has units of inverse time.
WORKED EXAMPLE
Compare two different loss rules
Two ideal stores start with 80 units. Store A loses 20 units per hour; store B loses one quarter of its current amount in each hour. How much remains after three hours?
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STOP & THINK
In dy/dt = −ky with constant positive k, what becomes smaller as a positive y decays?
- The magnitude of the rate |dy/dt|
- The fractional rate |(dy/dt)/y|
- The constant k
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Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Exponential change, logarithms and controlled approximations
Focus: Equal fractions, not equal amounts
Learning goals:
- Distinguish constant loss from constant fractional loss.
- Interpret a time constant and solve an exponential for elapsed time.
- Use dimensionless logarithms and test transformed graphs.
- Convert angles to radians and estimate approximation errors.
- Retain the leading nonzero term in a small difference.
Equations in this section (LaTeX):
\frac{dy}{dt}=-ky\quad\Longrightarrow\quad y(t)=y_0e^{-kt}\qquad(k>0)
Meaning: Differentiate the proposed solution to recover the rate equation. At t = 0 it returns the initial value y₀. The coefficient k has units of inverse time.
Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.
Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.02 / Exponential change, logarithms and controlled approximations
The time constant gives a useful comparison scale
Write the decay coefficient as k = 1/τ. The time constant τ is the time in which the quantity is multiplied by e⁻¹, approximately 0.368. It is not the time when the quantity becomes zero, and it is not generally the half-life. After two time constants the retained fraction is e⁻²; after three it is e⁻³.
The argument of an exponential must be dimensionless. You may write exp(−t/τ) because both t and τ are times. Writing exp(−t) without specifying an implicit time scale is physically incomplete. When you change from seconds to minutes, the numerical decay coefficient changes so that kt stays the same.
A positive exponent produces growth for positive time. A negative exponent produces decay toward zero. A quantity approaching a nonzero equilibrium often follows y = y∞ + (y₀ − y∞)exp(−t/τ): it is the difference from equilibrium that decays. Applying a logarithm directly to y in that situation can give a misleading straight-line test.
Follow equal time intervals along a decay
- Exponential decay
- 1
WORKED EXAMPLE
Change the clock without changing the process
A signal follows A = A₀ exp(−t/4 s). Rewrite its decay rate in inverse minutes and find the remaining fraction after 8 s.
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Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Exponential change, logarithms and controlled approximations Focus: The time constant gives a useful comparison scale Learning goals: - Distinguish constant loss from constant fractional loss. - Interpret a time constant and solve an exponential for elapsed time. - Use dimensionless logarithms and test transformed graphs. - Convert angles to radians and estimate approximation errors. - Retain the leading nonzero term in a small difference. Equations in this section (LaTeX): Build the physical reasoning before choosing an equation. Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
03 / Exponential change, logarithms and controlled approximations
A logarithm asks which exponent produced a ratio
If 10² = 100, then log₁₀(100) = 2. If eˣ = z, then ln z = x. A logarithm is the inverse of exponentiation with a chosen base. It is especially useful when the unknown appears in an exponent, because it brings that exponent down as a multiplier.
For positive real arguments, logarithms turn multiplication into addition and powers into multiplication. They do not turn a sum into a sum of logarithms: ln(a + b) is not ln a + ln b. Check that mistake using a = b = 1; ln 2 is nonzero, while ln 1 + ln 1 is zero.
The real logarithm requires a positive, dimensionless argument. In physics, ln(A/A₀) is meaningful because the units cancel. A graph labelled ln(A) usually has an implied reference unit; state that reference when interpreting its intercept. The logarithm of zero is not a finite number, and a negative signed coordinate cannot be inserted into a real logarithm merely because its unit is a metre.
| Expression | Equivalent expression | Use |
|---|---|---|
| ln(ab) | ln a + ln b | Products become sums |
| ln(a/b) | ln a − ln b | Ratios become differences |
| ln(aⁿ) | n ln a | An exponent becomes a multiplier |
| log₁₀ a | ln a / ln 10 | Change the base consistently |
WORKED EXAMPLE
Find when a signal reaches one quarter
For A/A₀ = exp(−t/4 s), find the time when the fraction is 1/4. Use ln 4 = 1.3863.
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STOP & THINK
A positive quantity is multiplied by 10³. By how much does its base-10 logarithm increase?
MEMBERSAnswer checking, feedback and the explanation are for members.Unlock →
Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Exponential change, logarithms and controlled approximations Focus: A logarithm asks which exponent produced a ratio Learning goals: - Distinguish constant loss from constant fractional loss. - Interpret a time constant and solve an exponential for elapsed time. - Use dimensionless logarithms and test transformed graphs. - Convert angles to radians and estimate approximation errors. - Retain the leading nonzero term in a small difference. Equations in this section (LaTeX): Build the physical reasoning before choosing an equation. Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
04 / Exponential change, logarithms and controlled approximations
Use ratios and transformed graphs to test a model
For an exponential with constant k, the ratio of two observations depends on their time separation, not on the unknown initial scale. This lets you estimate k even if the experiment started before your clock. Taking a logarithm of the ratio gives a direct expression for the rate coefficient.
A graph of ln(y/yref) against time is a straight line for a positive pure exponential: the slope is −k and the intercept depends on the chosen reference and time origin. A straight appearance is evidence to investigate, not proof of a unique mechanism. Additive measurement offsets, changing rates and a restricted data range can all mislead.
Uncertainty also changes under a transformation. For small relative uncertainty, the uncertainty in ln y is approximately Δy/y. Equal absolute errors in y therefore do not become equal vertical errors after taking logs. Small late-time signals can become especially uncertain relative to their size.
Both measured values must be positive and belong to the same constant-rate regime.
WORKED EXAMPLE
Find a half-time without knowing the starting signal
A positive signal falls from 48 units to 12 units in 6 s under a pure exponential model. How much time is needed for any signal value to halve?
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Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Exponential change, logarithms and controlled approximations
Focus: Use ratios and transformed graphs to test a model
Learning goals:
- Distinguish constant loss from constant fractional loss.
- Interpret a time constant and solve an exponential for elapsed time.
- Use dimensionless logarithms and test transformed graphs.
- Convert angles to radians and estimate approximation errors.
- Retain the leading nonzero term in a small difference.
Equations in this section (LaTeX):
\frac{y_2}{y_1}=e^{-k(t_2-t_1)},\qquad k=\frac{\ln(y_1/y_2)}{t_2-t_1}
Meaning: Both measured values must be positive and belong to the same constant-rate regime.
Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.
Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.05 / Exponential change, logarithms and controlled approximations
Why a small angle must be measured in radians
For a circle of radius r, define an angle in radians by θ = arc length/r. Because this ratio is dimensionless, a small sideways displacement can be compared directly with rθ. Degrees use a different numerical scale: 180 degrees correspond to π radians. Substituting a number of degrees into sin θ ≈ θ mixes incompatible numerical conventions.
Near zero, the sine curve has slope one when its argument is in radians. The tangent has the same leading slope. Their departures from θ appear at higher powers. The cosine starts at one and has zero initial slope; its first change is quadratic, not linear.
An approximation needs both a small parameter and a required accuracy. “Small” is not a fixed angle for every task. An angle acceptable for a rough geometric estimate may be unacceptable when a small energy difference is being obtained by subtracting nearly equal numbers.
These expansions use radians. Their first omitted terms estimate the leading errors when |θ| is sufficiently small.
WORKED EXAMPLE
Estimate the error before choosing an approximation
A geometric model replaces sin θ by θ at θ = 0.10 rad. Estimate the fractional error in the sine value using the leading omitted term.
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STOP & THINK
Which expression correctly uses the small-angle approximation for 5°?
- sin 5° ≈ 5
- sin 5° ≈ 5π/180
- sin 5° ≈ 180/(5π)
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Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Exponential change, logarithms and controlled approximations
Focus: Why a small angle must be measured in radians
Learning goals:
- Distinguish constant loss from constant fractional loss.
- Interpret a time constant and solve an exponential for elapsed time.
- Use dimensionless logarithms and test transformed graphs.
- Convert angles to radians and estimate approximation errors.
- Retain the leading nonzero term in a small difference.
Equations in this section (LaTeX):
\sin\theta=\theta-\frac{\theta^3}{6}+\cdots,\qquad\tan\theta=\theta+\frac{\theta^3}{3}+\cdots
Meaning: These expansions use radians. Their first omitted terms estimate the leading errors when |θ| is sufficiently small.
Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.
Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.06 / Exponential change, logarithms and controlled approximations
Keep the first term that survives the calculation
If you only need cos θ to rough absolute accuracy near zero, cos θ ≈ 1 can be useful. But suppose the required quantity is 1 − cos θ, as in the height gained by a pendulum. Replacing cos θ by one would give zero and erase the entire leading physical effect. You must retain the quadratic term.
This is a general lesson about subtracting nearby values. The error in either large value may be tiny relative to that value yet large relative to their small difference. Choose the expansion order after looking at the quantity you actually need.
For a simple pendulum at angle θ from the downward vertical, geometry gives h = L(1 − cos θ). At small angle, h ≈ Lθ²/2. The height is positive for either sign of θ, as a squared angle should predict. A linear height formula would give the wrong symmetry.
The constant terms cancel. The quadratic term becomes the leading contribution.
WORKED EXAMPLE
A small height that a rough cosine approximation would erase
A pendulum of length 0.80 m is displaced by 0.10 rad. Estimate the rise of its bob above its lowest point.
MEMBERSThe approach, step-by-step working and check are for members.Unlock →
Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Exponential change, logarithms and controlled approximations
Focus: Keep the first term that survives the calculation
Learning goals:
- Distinguish constant loss from constant fractional loss.
- Interpret a time constant and solve an exponential for elapsed time.
- Use dimensionless logarithms and test transformed graphs.
- Convert angles to radians and estimate approximation errors.
- Retain the leading nonzero term in a small difference.
Equations in this section (LaTeX):
\cos\theta=1-\frac{\theta^2}{2}+\cdots\quad\Longrightarrow\quad1-\cos\theta\approx\frac{\theta^2}{2}
Meaning: The constant terms cancel. The quadratic term becomes the leading contribution.
Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.
Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.07 / Exponential change, logarithms and controlled approximations
Approximate a model deliberately, then check its range
A related tool is the binomial approximation: (1 + ε)ⁿ ≈ 1 + nε when |ε| is small enough for the neglected terms to be acceptable. The leading omitted term is n(n − 1)ε²/2. A large exponent can make the error significant even when ε looks small; compare the actual omitted term with the accuracy you need.
The approximation explains why a small fractional radius change produces about three times that fractional volume change for a sphere. It also explains first-order sensitivity rules. It does not make finite percentage changes exactly additive, and it does not justify adding uncertainty bounds with signs as though independent deviations must cancel.
Before accepting any expansion, state its small parameter, keep the first nonzero contribution to the requested quantity, check dimensions and test an exact value if available. A limiting formula is a controlled simplification of a model. It cannot fix a physically inappropriate model.
WORKED EXAMPLE
First order and exact percentage changes
A sphere’s radius increases by 1% with no change in its shape. Compare the first-order and exact percentage increases in its volume.
MEMBERSThe approach, step-by-step working and check are for members.Unlock →
Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Exponential change, logarithms and controlled approximations Focus: Approximate a model deliberately, then check its range Learning goals: - Distinguish constant loss from constant fractional loss. - Interpret a time constant and solve an exponential for elapsed time. - Use dimensionless logarithms and test transformed graphs. - Convert angles to radians and estimate approximation errors. - Retain the leading nonzero term in a small difference. Equations in this section (LaTeX): Build the physical reasoning before choosing an equation. Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
BRING THE IDEAS TOGETHER
What should stay with you
- Exponential change has a constant fractional rate only within its model.
- Exponents and logarithm arguments must be dimensionless.
- A time constant, a half-time and a time to reach zero are different ideas.
- Small-angle expansions use radians.
- Choose an approximation order from the requested quantity and its accuracy.
RETRIEVE, THEN EXPLAIN
Try it without the examples
Work these out before checking. If an answer surprises you, identify which assumption or step needs another look. A correct choice is most useful when you can explain why the alternatives fail.
SELF CHECK 1
A pure exponential falls from 100 units to 25 units in 10 s. What is its half-time?
Answer in s.
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SELF CHECK 2
Using the small-angle approximation, by how many millimetres does a 1 m pendulum bob rise at an angle of 0.20 rad?
Answer in mm.
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SELF CHECK 3
Which is a valid real logarithm in a physical equation?
- ln(−3)
- ln(5 metres)
- ln(L/L₀), with positive lengths L and L₀
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SELF CHECK 4
Why does cos θ ≈ 1 fail when calculating the leading small-angle height L(1 − cos θ)?
- Because cosine is undefined near zero
- Because the constant terms cancel, making the quadratic term essential
- Because θ must always be measured in degrees
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APPLY THE CHAPTER
Take the ideas into a problem
Try a complete problem next. Commit to a method before revealing a hint, then compare your reasoning with the worked solution.
Recognize a constant fractional decreaseAdvanced · Hints and guided reasoningA capacitor that does not start emptyIntermediate · Hints and guided reasoningHalf the voltage, one quarter of the energyIntermediate · Hints and guided reasoningRead capacitance from a logarithmic graphIntermediate · Hints and guided reasoningHalve the excess twiceIntermediate · Hints and guided reasoningUse the slope to find a halving timeIntermediate · Hints and guided reasoningMake the exponent dimensionlessIntermediate · Hints and guided reasoningKeep the term that carries the energyIntermediate · Hints and guided reasoningEstimate an approximation’s errorIntermediate · Hints and guided reasoningCOME BACK TO THE IDEA
What can you explain later?
Try these from memory at your next study session, then again a few days later. Say your reasoning aloud or sketch it before looking. Adjust the gap to what you can explain.
Next study session
After one time constant of exponential decay, is the quantity zero? Explain what a time constant actually marks.
Compare your reasoning
It is 1/e of its initial value, about 37%. The time constant sets a multiplicative scale of change, not a finite stopping time.
Revisit the explanation →A few days later: change the situation
For a small angle θ in radians, when finding 1 − cos θ, why is replacing cos θ by 1 unhelpful?
Compare your reasoning
The leading terms cancel. You must retain the next term, giving 1 − cos θ approximately θ²/2. Choose the approximation order for the final quantity, not each expression in isolation.
Revisit the explanation →Compare the physical reason, the assumptions and the conclusion. Your wording can differ. If one part is missing, revisit that section and try explaining it again after a break.
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