Approximations & Logarithmic Scales
Scientific Notation — Large Numbers
Draw a line from each number in scientific notation to its standard form.
Scientific Notation — Small Numbers
Draw a line from each number in scientific notation to its standard form.
Convert to Scientific Notation
Circle the correct scientific notation for each number.
45,000
0.0072
6,300,000
Powers of 10
Circle the correct value for each power of 10.
10³
10⁻²
10⁰
10⁻⁴
Order of Magnitude
Sort each quantity by its order of magnitude: Small (10⁻⁶ or less), Medium (10⁻⁵ to 10⁵), or Large (10⁶ or more).
Match Quantity to Estimate
Draw a line from each quantity to its approximate order of magnitude.
Reasonable Estimates — Everyday
Circle the most reasonable estimate for each quantity.
Mass of a car
Height of Mt Everest
Diameter of a coin
Reasonable Estimates — Science
Circle the most reasonable estimate for each scientific quantity.
Distance from Earth to the Sun
Mass of a grain of sand
Width of a human hair
Which Rounding Is Best?
Circle the most appropriate level of rounding for each scenario.
Estimating how many seats in a stadium
Dosage of medicine for a patient
National population for a news report
Cutting timber for a bookshelf
Rounding Errors — Single Calculation
Circle the correct answer.
π rounded to 2 decimal places is 3.14. The true value starts 3.14159. The absolute error of rounding is approximately:
If you round 2.745 to 2 decimal places, you get:
The percentage error when using 3.14 for π is approximately:
Accumulated Rounding Errors
Circle the correct statement about how rounding errors behave.
When you multiply a rounded number by itself many times, the error:
Using π ≈ 3.14 vs π ≈ 3.14159 for the area of a circle with r = 1:
In financial calculations, rounding to the nearest cent after each step instead of at the end:
Match the Rounding Strategy
Draw a line from each scenario to the best rounding strategy.
Rounding Error Investigation
Investigate how rounding affects repeated calculations.
Calculate 1.1⁵ using (a) the exact value 1.1, and (b) the rounded value 1. What is the difference? Now try 1.1¹⁰. Explain what happens to the error as the power increases.
A bank calculates daily interest at 0.0137% (≈ 5% annual). Compare the result of rounding to 2 decimal places each day vs rounding only at the end of the year, starting with $10,000. Which gives a more accurate result?
Logarithmic Scales in Real Life
Draw a line from each logarithmic scale to what it measures.
Richter Scale Questions
Circle the correct answer about the Richter scale.
Each whole number increase on the Richter scale means the amplitude is:
Each whole number increase means roughly how much more energy?
A magnitude 7 earthquake has how many times more amplitude than magnitude 5?
A magnitude 8 earthquake releases how many times more energy than magnitude 6?
Decibel Scale Questions
Circle the correct answer about the decibel scale.
An increase of 10 dB means sound intensity is:
A whisper is ~30 dB and a vacuum cleaner is ~70 dB. The vacuum is how many times more intense?
A sound at 80 dB compared to 60 dB is:
pH Scale Questions
Circle the correct answer about the pH scale.
Each pH unit represents a factor of:
A solution with pH 3 is how many times more acidic than pH 5?
Pure water has pH 7. Lemon juice has pH 2. The juice is:
Linear vs Logarithmic Relationships
Sort each relationship: Linear Scale or Logarithmic Scale.
Comparing Earthquakes
Use the Richter scale to compare earthquakes. Show your reasoning.
The 2011 Christchurch earthquake was magnitude 6.2 and the 2011 Tōhoku (Japan) earthquake was magnitude 9.1. How many times more amplitude did the Japan earthquake have? How many times more energy? Show your calculations.
If a magnitude 4 earthquake is barely felt, explain why a magnitude 6 earthquake (only 2 more on the scale) can cause major damage. Use the energy multiplier to support your explanation.
Decibel Problem Solving
Apply your understanding of the decibel scale.
A conversation is 60 dB. A rock concert is 110 dB. How many times more intense is the concert than a conversation? If prolonged exposure above 85 dB causes hearing damage, how many times more intense is the damage threshold than normal conversation?
Two speakers each produce 70 dB. When both play together, the combined level is about 73 dB, NOT 140 dB. Explain why doubling the number of identical sound sources only adds about 3 dB.
Approximation in Repeated Calculations
Investigate how approximation errors compound.
A calculator shows π as 3.14. Another uses 3.14159265. Calculate π¹⁰ using each value. What is the absolute and percentage difference? Explain why repeated multiplication amplifies small initial errors.
A GPS has a 3-metre accuracy for each position fix. If a bushwalker takes 50 position readings on a hike, explain why the total distance error could be much more than 3 m. Under what conditions would errors cancel out vs accumulate?
Create Your Own Logarithmic Scale
Design a logarithmic scale for a real-world phenomenon.
Social media posts can get anywhere from 1 view to 1 billion views. Design a 'Virality Scale' from 1 to 10 where each level represents 10× more views. Define each level (1 = 1–10 views, etc.) and give an example of what type of post might reach each level.
True or False — Logarithmic Scales
Circle TRUE or FALSE for each statement.
On a logarithmic scale, equal distances represent equal ratios
A magnitude 8 earthquake is twice as strong as magnitude 4
The difference between pH 3 and pH 4 represents the same factor as between pH 8 and pH 9
Logarithmic scales are useful when data spans many orders of magnitude
Significant Figures & Precision
Explain the role of significant figures in maintaining appropriate precision.
A student measures a rectangle as 12.3 cm × 4.56 cm and reports the area as 56.088 cm². Explain why this answer implies false precision. What should the answer be, and why?
Explain the difference between accuracy and precision using an example. Why is it misleading to report a very precise answer from inaccurate measurements?
Logarithmic Scales in Daily Life
Explore logarithmic scales you encounter in everyday life.
- 1Find the decibel levels of 5 common sounds (whisper, conversation, lawnmower, concert, jet engine). Calculate how many times more intense each is than the quietest.
- 2Look up the pH of common household substances (lemon juice, water, soap, bleach). Calculate the hydrogen ion concentration factor between each.
- 3Research the Richter scale magnitudes of 3 famous Australian earthquakes. Calculate the energy differences between them.
Approximation Error Experiment
Test how rounding affects calculations in practice.
- 1Using a calculator, compute (1 + 0.001)¹⁰⁰⁰ with full precision, then try rounding 1.001 to 1.00 first. How different are the results? This models how tiny interest rates compound.
- 2Measure a square tile to the nearest cm, then to the nearest mm. Calculate the area both ways. How much does the extra precision matter?
- 3Find an online compound interest calculator. Compare results when the rate is entered as 5% vs 5.00% vs 4.99%. How much difference does the small rounding make over 30 years?
Logarithms — Basic Evaluation
Evaluate each logarithm and explain your reasoning.
Evaluate: (a) log₂(8) (b) log₃(81) (c) log₁₀(10000) (d) log₅(125). Show the exponential equation that matches each.
Evaluate: (a) log₂(1/4) (b) log₁₀(0.001) (c) log₃(1). Explain why logarithms of fractions less than 1 give negative results.
Logarithm to Exponential Form
Draw a line from each logarithmic statement to its equivalent exponential form.
Logarithm Laws — Which Law?
Circle the logarithm law used in each step.
log(6) = log(2) + log(3)
log(50) = log(100) − log(2)
log(8) = 3·log(2)
log(1) = 0 for any valid base
Classify: Exact or Approximate Logarithm
Sort each logarithm value: Exact Integer or Approximate Decimal.
Scientific Notation — Operations
Perform calculations in scientific notation and express answers correctly.
Calculate (3.0 × 10⁴) × (2.5 × 10³). Express your answer in correct scientific notation and standard form.
Calculate (9.6 × 10⁸) ÷ (3.2 × 10⁵). Express your answer in correct scientific notation.
Add: (4.5 × 10⁶) + (2.3 × 10⁵). Why must you adjust exponents before adding? Show the working.
Error Bounds — Upper and Lower
Circle the correct upper and lower bounds for each measurement.
A length measured as 7.4 cm (to 1 dp). The bounds are:
A mass rounded to 3 kg. The bounds are:
A time of 12 s rounded to the nearest second. The bounds are:
Logarithmic Scale Design Challenge
Apply logarithmic thinking to a novel context.
Design a 'Speed Scale' for vehicles (bicycle to rocket). Choose 5 speeds ranging from ~5 km/h to ~40,000 km/h. Create a log₁₀ scale and mark each vehicle. What is the advantage of this scale over a linear one?
Logarithmic Scale Values — Order
Put these values in order from smallest to largest.
Compound Approximation Errors
Analyse how small approximation errors compound in multi-step problems.
A student estimates the volume of a sphere using r ≈ 5 cm instead of the true 5.2 cm. Calculate the volume using both values (V = 4/3 × π × r³). What is the absolute error? What is the percentage error?
If you use g ≈ 10 m/s² instead of 9.8 m/s² in physics calculations, what is the percentage error? For a projectile with initial velocity 20 m/s, compare the range using each value: R = v²/g.
Logarithm Laws — Simplify
Draw a line from each logarithm expression to its simplified form.
Sound Levels Survey
Measure and compare real sound levels using the decibel scale.
- 1Download a free sound level meter app. Measure dB levels in 5 different locations (quiet room, outside, kitchen, TV, road). Calculate how many times more intense the loudest is compared to the quietest using 10^((dB₁−dB₂)/10).
- 2Record the sounds in your house over 10 minutes. Make a tally of time spent above 70 dB (busy street level). Research what sustained exposure to those levels means for hearing health.
- 3Research: what dB level does a smoke alarm produce? A normal conversation? A classroom? Create a personal 'sound diary' for one school day.
Interpreting Logarithmic Graphs
Read and interpret a graph drawn on a logarithmic scale.
A graph of bacterial growth is plotted with time (hours) on the x-axis and number of bacteria on a log₁₀ y-axis. At t=0 the value is 2 (meaning 100 bacteria). At t=4 the value is 5 (meaning 100,000 bacteria). Calculate the growth rate per hour and express the population as an exponential function.
On a log₁₀ scale, the distance between the marks for 10 and 100 equals the distance between 100 and 1,000. Explain why this is the case in terms of what the scale is actually measuring.
Scientific Notation Practice Errors
Tally common errors students make with scientific notation.
| Item | Tally | Total |
|---|---|---|
Wrong power of 10 selected | ||
Coefficient not between 1–10 | ||
Wrong sign on exponent | ||
Arithmetic error on exponent | ||
Forgot to adjust coefficient | ||
Confused multiply vs divide |
Orders of Magnitude — Estimation
Use orders of magnitude to make rough estimates.
Estimate the number of heartbeats in an average human lifetime. Show your assumptions and reasoning, expressing your final answer in scientific notation.
Estimate the number of words in all the books in a typical school library. State your assumptions clearly and express your answer to the nearest order of magnitude.
Significant Figures — Count Correctly
Circle the number of significant figures in each value.
0.00420
1.050 × 10⁴
3700 (measured to nearest 100)
0.0306
Real-World Logarithm Application
Apply logarithm properties to solve a real-world problem.
The Richter magnitude M = log₁₀(A/A₀). If one earthquake has amplitude 500 times the reference, calculate M. If a second has amplitude 50,000 times the reference, calculate its M. How many times stronger is the second earthquake in terms of amplitude?
Accuracy in Everyday Measurement
Investigate how measurement precision affects everyday decisions.
- 1Measure the same object (e.g. a book or door) using a ruler to the nearest cm, then to the nearest mm. Calculate the area or volume using each measurement. How much does the extra precision change the result?
- 2When cooking, many recipes say '1 cup' of flour. Weigh out what you think is 1 cup, then look up the exact mass (approx. 125 g). Calculate your percentage error.
- 3Find an example of significant figures in a science or maths textbook. Identify: (a) how many significant figures are in the input data, and (b) how many should be in the answer.
Classify the Error Type
Sort each error: Rounding Error, Truncation Error, or Measurement Error.
Significant Figure Rules — Order of Steps
Put the steps in order for reporting a calculation result to the correct number of significant figures.
Fermi Estimation — Orders of Magnitude
Use order-of-magnitude reasoning to estimate large quantities.
How many piano tuners are in Melbourne? Break this into: population, fraction who own pianos, how often pianos are tuned, how many pianos a tuner can tune per year. Show all steps and give your answer as an order of magnitude.
pH Values to Substance
Draw a line from each substance to its approximate pH value.
Using Logarithms to Solve Exponential Equations
Circle the correct method or answer for each equation.
To solve 10ˣ = 500, take:
log₁₀(x) = 2.5 means x =
log₂(x) = 6 means x =
Logarithm Change of Base
Apply the change of base formula log_b(x) = log(x) / log(b).
Calculate log₇(100) using the change of base formula and a calculator. Show your working.
Use the change of base formula to explain why log₁₀(x) and ln(x) are simply multiples of each other (they differ by the constant factor log₁₀(e) ≈ 0.4343).
Logarithm Applications — Field
Sort each application of logarithms into the correct field.
Scientific Notation in Astronomy
Apply scientific notation to astronomical distances and scales.
The distance from Earth to Alpha Centauri is about 4.13 × 10¹³ km. Light travels at 3.0 × 10⁵ km/s. Calculate how many seconds light takes to reach Alpha Centauri. Convert to years (1 year ≈ 3.15 × 10⁷ s). What does this tell you about the distances involved?
Logarithm Evaluation — Mental Math
Evaluate each logarithm without a calculator.
log₁₀(10,000,000)
log₂(1/8)
log₁₆(4)
log₁₀(0.0001)
Logarithmic Thinking — Growth Rate
Use logarithms to find when exponential processes reach a target.
A city's population grows at 3% per year. Currently 500,000 people. Using P = 500,000 × 1.03ⁿ and taking log₁₀ of both sides, find when the population reaches 1,000,000. (Hint: log₁₀(2) ≈ 0.301, log₁₀(1.03) ≈ 0.0128)
Approximation Errors in Navigation
Explore how small measurement errors compound in navigation.
- 1Research the 'dead reckoning' navigation method used historically on ships. If a ship's speed is estimated with 2% error and direction with 1° error, how far off-course could the ship be after 500 km? Use trigonometry and proportion to estimate.
- 2Look up how GPS satellites achieve their accuracy. What happens to positional error if the satellite clock is out by 1 microsecond? (Hint: light travels 300 m in 1 microsecond.)
- 3Research the Mars Climate Orbiter crash of 1999 (caused by a units error: metric vs imperial). Calculate how large the navigation error was and what small measurement mistake caused it.
Richter Scale — Deep Dive
Apply logarithmic reasoning to compare earthquake energies in depth.
The energy E (in joules) released by an earthquake of magnitude M is approximated by log₁₀(E) = 4.8 + 1.5M. (a) Find the energy of an M = 5 earthquake. (b) Find the energy of an M = 7 earthquake. (c) How many times more energy does M = 7 release than M = 5? (d) Explain why the energy ratio grows so fast compared to the magnitude difference.
Logarithm Practice — Types of Problems
Tally the types of logarithm problems you solve in a practice session.
| Item | Tally | Total |
|---|---|---|
Evaluating simple logarithms | ||
Applying logarithm laws | ||
Solving exponential equations using logs | ||
Real-world log scale problems | ||
Change of base calculations |
Approximation vs Exact — When Does It Matter?
Analyse when exact values are needed vs approximations.
A structural engineer calculates the load-bearing capacity of a beam as exactly 12,450 kg. A colleague rounds this to 12,000 kg for convenience. Discuss: (a) In what contexts is this rounding acceptable? (b) In what contexts could it be dangerous? (c) What is the percentage error?
In mathematics, we often use exact surd values (e.g. √2 = 1.41421356...). Explain why a computer programmer might prefer an approximation like 1.4142 and when this creates problems in software (e.g. comparing floating-point numbers for equality).
Solving 10ˣ = N Using Logarithms — Steps
Put the steps in the correct order for solving 10ˣ = N.
Natural Logarithm — Introduction
Introduce the natural logarithm ln(x) and its connection to e.
The number e ≈ 2.71828 is the base of natural logarithms. Calculate: (a) ln(e) — the answer should be 1. (b) ln(e²). (c) ln(e⁻³). (d) e^(ln 5). Explain the pattern: what is the relationship between eˣ and ln(x)?
Logarithm Laws — Identify the Law Used
Sort each equation into the logarithm law it demonstrates.
Solving Logarithmic Equations
Solve equations involving logarithms by applying log laws.
Solve: log₂(x) = 5. Show your steps and verify your answer.
Solve: log(x) + log(x−3) = 1. Explain any restrictions on x.
Solve: 3^x = 20 by taking logarithms of both sides. Give your answer to 3 decimal places.
Bacteria N = 500 × 3^t. When will the count first exceed 50,000? Show full working.
Applications of Logarithmic Scales
Interpret and compare real-world data on logarithmic scales.
The Richter scale is logarithmic. A magnitude 6.0 earthquake releases 31.6× more energy than magnitude 5.0. How much more energy does magnitude 7.0 release than 5.0?
Sound is measured in decibels on a log scale. A whisper is 30 dB, conversation 60 dB. How many times more intense is conversation?
Why are logarithmic scales used for data spanning many orders of magnitude? Give two advantages.