Transform and interpret a distribution on a base-10 log scale

Quantities or scale rule

\(\displaystyle 1,10,100,1000\)

A sequence of masses is 1, 10, 100, 1000 micrograms. What factor takes each mass to the next?

Would equal distances on an ordinary linear scale fit all four masses equally spaced?

Quantities or scale rule

\(\displaystyle 1,10,100,\ ? \)

A logarithmic mass scale has equally spaced ticks 1, 10, 100 and then an unlabelled tick. What mass belongs at the next tick, in micrograms?

Move the point by whole decades, then half decades. Compare its movement on the two scales. Switch to the small-quantity range and repeat. The two scales show the same quantity, with their different starting values stated below

What stays constant when the point moves one whole decade on the logarithmic scale?

A logarithmic scale uses multiplicative steps

On a base-10 logarithmic scale, each equal decade step multiplies the quantity by ten. The tick labels can show the original quantities. Position is proportional to their exponents, not to the quantities themselves

1. 1Read the labels and units A tick labelled 100 micrograms means 100 micrograms, even though its base-10 logarithm is 2.

Quantities or scale rule

\(\displaystyle 10\text{ to }1000\)

A particle mass rises from 10 to 1000 micrograms. How many times the original mass is the new mass?

Quantities or scale rule

\(\displaystyle 1000\text{ to }10\)

A mass falls from 1000 to 10 micrograms. What fraction of the original mass remains?

A graph rises by two base-10 log units. Which statement is justified?

Quantities or scale rule

\(\displaystyle 10^{-6},10^{-5},10^{-4},10^{-3}\)

A concentration scale has ticks 0.000001, 0.00001, 0.0001 and 0.001 mol/L. What factor takes each concentration to the next?

Quantities or scale rule

\(\displaystyle 10^{-3}\)

What ordinary decimal is represented by 10 to the power −3?

Does a negative exponent on this scale mean a negative concentration?

Where should a zero measurement be placed on an ordinary logarithmic scale?

Quantities or scale rule

\(\displaystyle 10\text{ and }1000\)

On a logarithmic mass scale, 10 and 1000 micrograms are two decades apart. What mass is halfway between them?

Someone places 505 micrograms halfway between 10 and 1000 on a logarithmic scale. What went wrong?

A laboratory uses a logarithmic scale to display masses spanning six powers of ten. What does this change?

Two reports use different minimum values and different plot widths. Can you compare their physical distances on the page directly?

Which operation moves a positive raw value to its base-10 log coordinate?

Base-10 logarithmic scales

A base-10 log scale is useful when positive values cover several orders of magnitude. Apply log base 10 to transform a raw value, and use a power of 10 to return to the raw scale

1. 1Transform a positive value Apply log base 10 to each positive raw value when moving from the raw scale to the log scale \(\displaystyle \log_{10}(1000)=3\)
2. 2Interpret a log coordinate A log coordinate b represents the raw value 10^b, not the number b itself \(\displaystyle \log_{10}(x)=2\quad\Rightarrow\quad x=10^2=100\)
3. 3Read equal log widths Equal widths on a base-10 log scale represent equal multiplicative factors on the raw scale \(\displaystyle \log_{10}(100)-\log_{10}(10)=1\quad\Rightarrow\quad\frac{100}{10}=10\)
4. 4Compare transformed shapes Describe a transformed distribution using its log coordinates, because the transformation can change the apparent shape \(\displaystyle \log_{10}(10x)=1+\log_{10}(x)\)

Data or rule

\(\displaystyle \log_{10}(1000)\)

Find the base-10 log coordinate of 1000.

Data or rule

\(\displaystyle \log_{10}(x)=3\)

Find the raw value represented by the log coordinate 3.

Reading a log coordinate as a raw value. One of these lines is right. Which one?

Enter the operation and preserve precision

On a calculator or spreadsheet choose base-10 LOG, not natural logarithm LN. Use 10 raised to a power for the reverse operation. Keep the unrounded value until the final requested answer

1. 1Check magnitude first Since 100<250<1000, log₁₀(250) is between two and three. Since 10²<10^2.4<10³, its antilogarithm is between 100 and 1000.
2. 2Check by the inverse Exponentiating an unrounded base-10 logarithm should recover the positive input. Exponentiating a rounded logarithm usually gives only an approximation

Data or rule

\(\displaystyle \log_{10}(250)\)

Use technology to evaluate this logarithm, then give the result to three decimal places

Data or rule

\(\displaystyle \log_{10}(N)=2.4\)

Use technology to reverse the logarithm, giving N to three decimal places

State what the logarithm measures

Use positive measurements of the same quantity in the same unit. A precise dimensionless label is log₁₀(q/u), where u is the common reference unit. A log axis can show equally spaced powers of ten in the original units; an axis explicitly labelled log₁₀(q/u) instead shows their exponents

1. 1Equal intervals mean equal factors The positions of 1, 10, 100 and 1000 in the same unit are equally spaced. Equal arithmetic increases are generally not equally spaced
2. 2Respect the order of comparison For A/B subtract log(B) from log(A), even if the difference is negative. Reverse the order to obtain the reciprocal ratio

One mass is logged in grams and another in kilograms. What must happen before subtracting their log values to compare masses?

Construct a log-scale representation

Four positive measurements are 0.1, 1, 10 and 100 metres. Draw a labelled base-10 log axis with equally spaced decade ticks, place each measurement, and also state its coordinate on an axis labelled log₁₀(q/1 m). Add the point √10 metres. Compare the largest with the smallest, explain the spacing and say whether zero metres can be plotted

1. 1Write before continuing Produce your complete table, diagram or explanation independently. The next screen is a model for self-review; the complete written response is not automatically marked

Model response: Construct a log-scale representation

Compare your working, labelled axes, numerical checks and conclusion with this model

1. 1Coordinates and labels The four exponent coordinates are −1, 0, 1 and 2, so equally spaced ticks can be labelled 0.1, 1, 10 and 100 m. Alternatively label the axis log₁₀(q/1 m) with ticks −1, 0, 1 and 2. Do not label the exponent values as metres
2. 2Intermediate point and ratio √10 m has exponent 1/2, halfway between 1 m and 10 m. The log difference from 0.1 m to 100 m is three, so the ratio is 10³=1000. Each equal interval multiplies the original measurement by ten
3. 3Domain Zero has no finite real base-10 logarithm and cannot occupy a point on this axis. Positive values below the unit have negative exponent coordinates, not negative original measurements

Build a distribution in the stated coordinates

Transform each strictly positive observation, preserve duplicates and sample size, and then count it in exactly one stated interval. Zero and negative original measurements have no ordinary real base-10 logarithm; do not silently replace, delete or shift them

1. 1Name the coordinate For a measurement m in milligrams, use y=log₁₀(m/1 mg). The ratio inside the logarithm is dimensionless. A y-axis label on the horizontal coordinate shows exponents; a logarithmic axis labelled in milligrams instead shows the original positive quantities
2. 2Map intervals monotonically Because base ten exceeds one, a≤y<b corresponds to 10ᵃ≤m/(1 mg)<10ᵇ. Keep each endpoint inequality. An equal log width w represents raw endpoint ratio 10ʷ, not raw difference w.
3. 3Keep boundaries and precision Use the specified convention: these examples include each lower boundary and exclude each upper boundary, except the final upper endpoint is included. Classify using unrounded logarithms or equivalent exact raw boundaries; rounding can move a value into the wrong class
4. 4Do not assume improved shape A transformation changes numerical distances and can change the displayed shape. Inspect both distributions with stated bins. A symmetric transformed histogram alone does not establish a normal model, and a log transform need not make a distribution more symmetric

Requested interval

\(\displaystyle 0\leq y<2\)

How many observations have log coordinates from 0 inclusive to 2 exclusive?

Data or rule

\(\displaystyle n=9,\quad f=5\)

Five of nine observations satisfy the requested interval. Give the percentage to one decimal place

Data or rule

\(\displaystyle b-a=0.5\)

A log-coordinate interval has width 0.5. What multiplicative raw endpoint factor does it represent? Give two decimal places

Coordinate and bin convention

\(\displaystyle y=2,\quad [1,2),\ [2,3),\ [3,4]\)

A positive raw value x=100 has y=log₁₀(x)=2. With bins [1,2), [2,3), [3,4], where does it belong?

Log-coordinate class frequencies

\(\displaystyle 1,2,3,2,1\)

The same nine fictional particle masses produce this raw-scale histogram. Their log-coordinate histogram has frequencies 1,2,3,2,1 in five equal-width classes. What is supported?

Produce and verify both distributions

Use these nine fictional particle masses, in milligrams: 0.1, 1, 1, 10, 10, 10, 100, 100, 1000. Produce a table of every raw value and y=log₁₀(m/1 mg), preserving duplicates. Construct a raw histogram with boundaries 0,250,500,750,1000,1250 mg and a transformed histogram with boundaries −1,0,1,2,3,4. Include lower endpoints, exclude upper endpoints except the final upper endpoint, and label both axes

1. 1Interpret an interval Find the number and percentage with 1≤m<100 mg, reporting the percentage to one decimal place. State the matching log interval and explain why the raw value at its upper boundary is excluded. State the factor associated with one log class
2. 2Compare and evaluate Compare the two shapes, explain what changes and what remains invariant, and give a limit on any normal-model claim. Discuss why the two histograms use different horizontal units and why zero on the raw horizontal axis is not a valid original value for the log transform
3. 3Choose and verify technology Choose spreadsheet or statistical functionality and explain why it fits the task. Record inputs, base-10 operation, reference unit, bin boundaries, inequalities, windows and rounding. Check every transform using an inverse operation and verify all bin counts manually against the original values
4. 4Boundary counterexamples Without adding observations to the supplied sample, state where a hypothetical 10000 mg value would belong in the stated log bins, and explain how prematurely rounding log₁₀(9.99) to two decimal places could misclassify 9.99 mg. Explain how you would investigate a zero or negative original record
5. 5Complete before continuing Save the table, two histograms, calculations, explanation and tool record before the model. Complete plotting, writing and external-tool execution are not automatically marked; use self or teacher review

Model response: preserve records while changing coordinates

The raw/log pairs are (0.1,−1), (1,0), (1,0), (10,1), (10,1), (10,1), (100,2), (100,2), (1000,3). Both displays represent nine observations. The ordered observations and duplicate counts are preserved; additive distances, units and histogram bins change

1. 1Complete displays For raw intervals [0,250),[250,500),[500,750),[750,1000),[1000,1250] mg, use frequencies 8,0,0,0,1. The 1000 mg record belongs in the last class. For log intervals [−1,0),[0,1),[1,2),[2,3),[3,4], use frequencies 1,2,3,2,1. Draw touching equal-width columns within each coordinate system, with frequency vertically. Suitable windows are raw 0–1250 mg by 0–9 and log −1–4 by 0–4. Label the log axis y=log₁₀(m/1 mg), not milligrams
2. 2Interval and ratio The raw interval 1≤m<100 mg corresponds to 0≤y<2. Count the two 1s and three 10s: five of nine, or 55.6% to one decimal place. The two 100s are excluded. The raw upper boundary represented by y=2 is 100 mg. Each log class has width one and endpoint ratio ten, but raw widths differ: [1,10) is 9 mg wide and [10,100) is 90 mg wide
3. 3Shape and limits The specified raw histogram has a long high-side tail. The transformed histogram’s heights are symmetric about its central class. This supports describing these binned sample displays; it does not establish that an underlying population is normal or that every log transform improves symmetry. The raw axis may begin at zero, but an actual zero original observation has no finite log coordinate
4. 4One reproducible workflow Enter the nine raw masses in A2:A10 and use LOG10(A2) in B2, copied down, treating the numerical milligram value as m/(1 mg). Keep stored logs unrounded. Check 10^B2 against A2 for every row. Build both frequency tables with COUNTIFS >=lower and <upper, changing only the last upper test to <=. Check sums of nine. Plot touching equal-width columns with the stated numeric boundaries, not unlabelled category bars. Verify each original record independently and document the exact functions/settings and choice rationale
5. 5Counterexamples and data handling A hypothetical 10000 mg value has y=4 and belongs to the final closed interval [3,4]; it was not added to the nine records. For 9.99 mg, log₁₀(9.99)≈0.99956549, which belongs to [0,1). Rounding first to 1.00 would wrongly assign [1,2). Use unrounded values or exact raw boundaries. Investigate a zero/negative record’s source, units and meaning; do not silently delete it or add a constant, since that changes the data or transformation

Stored and displayed coordinate

\(\displaystyle y=0.99956549\ldots,\quad \text{displayed to 2 d.p.: }1.00\)

For the stated bins [0,1) and [1,2), how should a positive 9.99 mg observation be classified after y=log₁₀(m/1 mg)?