📘 1. Scaling Very Large or Small Numbers Using Logarithms
- Purpose : To compress large numerical ranges into manageable scales so data can be represented and compared easily.
- Example : Earthquake magnitude (M) = log 10(I / I0)
— Each increase of 1 in M represents 10× increase in intensity. - Advantages :
- Highlights rate of change rather than absolute values.
- Useful for data ranging over many orders of magnitude (e.g., 10−6 to 109).
Worked Example :
Suppose a quantity increases from 5 to 5 000 000.
On a log10 scale this becomes log(5 000 000) − log(5) = 6 − 0.7 ≈ 5.3 orders of magnitude.
This shows the growth rate clearly without the numbers becoming unmanageable.
🌍 Real-World Connections
- Biology : Bacterial growth and enzyme kinetics use semi-log plots to analyze rates and half-lives.
- Chemistry : Activation energy from Arrhenius equation (ln k vs 1 / T).
- Economics : Log-log graphs used to compare income vs consumption (power law).
- Social Media : Follower growth modeled using logarithmic scaling to interpret relative growth rates.
📐 IA Spotlight
- Use real datasets (e.g., GDP growth, radioactive decay, population growth) and apply logarithmic scaling to linearize relationships.
- Compare log-log and semi-log fits to decide whether data follows a power law or exponential trend.
- Interpret slope and intercept in context and verify model validity using correlation coefficients.
📗 2. Linearizing Data Using Logarithms
Logarithms can convert non-linear data to a linear form so that parameters can be determined using a best-fit straight line.
- Exponential Relationship : y = A ek x
Taking natural logs: ln y = ln A + k x → straight line of form y = m x + c.
Slope = k, Intercept = ln A. - Power Relationship : y = A xn
Taking logs of both sides: log y = n log x + log A.
Slope = n, Intercept = log A.
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Worked Example :
Suppose the data (x, y): (1, 2), (2, 4), (3, 9), (4, 16).
Take log10 of both x and y:
| x | y | log x | log y |
|---|---|---|---|
| 1 | 2 | 0 | 0.30 |
| 2 | 4 | 0.30 | 0.60 |
| 3 | 9 | 0.48 | 0.95 |
| 4 | 16 | 0.60 | 1.20 |
Plot log y against log x → straight line with gradient ≈ 2 ⇒ y = A x2.
🔍 TOK Perspective
When we apply logarithms to data, we change how we perceive relationships — a straight line appears only after a non-linear transformation.
Does this make mathematics a tool of discovery or a lens of interpretation?
Are we revealing truths about the world or constructing them through mathematical frameworks?
❤️ CAS Ideas
- Conduct an experiment tracking plant growth or decay and analyze data using logarithmic plots.
- Develop an educational infographic explaining logarithmic scales for younger students (e.g., pH scale, decibels).
📙 3. Interpreting Log-Log and Semi-Log Graphs
- Semi-log graph : One axis (logarithmic), the other linear → used for exponential data.
- Log-log graph : Both axes (logarithmic) → used for power law data.
- Slope of log-log graph : Represents exponent n in y = A xn.
- Slope of semi-log graph : Represents rate constant k in y = A ek x.
Example : If log y vs x is a straight line with slope −0.3, then k = −0.3 and y = A e−0.3 x.
🌐 EE Focus
- Investigate applications of logarithmic linearization in scientific data analysis (e.g., Arrhenius plots, spectroscopy).
- Explore the validity of power laws in economics (Pareto distribution, wealth inequality).
- Compare natural and base-10 log scales and their historical evolution in mathematical modeling.
📝 Paper 1 & 2 Exam Tips
- Recognize when to use logarithmic transformation to linearize data.
- Interpret gradients and intercepts clearly in context of the model.
- Remember the difference between log-log and semi-log plots in determining relationships.
- Use GDC to find logarithmic regression fits and display best-fit lines efficiently.
🧠 Examiner Tip
Students often forget to state the base of the logarithm used. Always indicate whether it is log10 or ln.
On logarithmic graphs, label axes clearly and state the relationship deduced from the linear form.
Full marks require interpretation of the slope and constant in context.