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A difference of 1 on a base −10 logarithmic scale represents a tenfold ratio.
Worked example 1
01 / Three hypothetical object masses, in kilograms, are 1×10−5, 2×10−1, 3×103. Tabulate their base −10log coordinates using 1 kilogram as reference. Plot the three masses on an ordinary scale and on a logarithmic coordinate scale, explain why the latter separates small values, and find the log− coordinate change for multiplying a mass by 10. In an equal-ratio pitch model, twelve equal steps double a frequency. Derive the frequency ratio for one step and find the frequency three steps above 180 hertz. Do not interpret negative log coordinates as negative mass. Show written work; selecting a summary alone is not a complete response. Complete the written work or drawing first. The selected answer checks the result only; your reasoning or drawing needs teacher/self-review.
02 / L=log10(1kgW) is dimensionless and requires W>0. In the selected unit, log coordinates are −5, −0.69897, 3.477121.
03 /
Mass ( kg )
Log coordinate
1×10−5
−5
2×10−1
−0.69897
3×103
3.477121
04 / log10(qW)−log10(W)=log10q. Here the change is 1. Equal additive log steps represent equal multiplicative physical factors. Negative log values mean a positive mass smaller than the reference unit.
05 / Ordinary mass scale. Open circles: observations or key values.
06 / Logarithmic coordinate scale. Open circles: observations or key values.
07 / q12=2⇒q=2121. Three steps give f=180q3=180241≈214.0573 hertz. This illustrates the logarithmic pitch premise only, not an external music-tuning project.
Worked example 2 · Interpret logarithmic coordinates and equal ratios
Three hypothetical object masses, in kilograms, are 1.1×10−6, 2.2×10−1, 3.1×103. Tabulate their base −10log coordinates using 1 kilogram as reference. Plot the three masses on an ordinary scale and on a logarithmic coordinate scale, explain why the latter separates small values, and find the log− coordinate change for multiplying a mass by 15.848932. In an equal-ratio pitch model, twelve equal steps double a frequency. Derive the frequency ratio for one step and find the frequency three steps above 185 hertz. Do not interpret negative log coordinates as negative mass. Show written work; selecting a summary alone is not a complete response. Complete the written work or drawing first. The selected answer checks the result only; your reasoning or drawing needs teacher/self-review.
L=log10(1kgW) is dimensionless and requires W>0. In the selected unit, log coordinates are −5.958607, −0.657577, 3.491362.
Mass ( kg )
Log coordinate
1.1×10−6
−5.958607
2.2×10−1
−0.657577
3.1×103
3.491362
log10(qW)−log10(W)=log10q. Here the change is 1.2. Equal additive log steps represent equal multiplicative physical factors. Negative log values mean a positive mass smaller than the reference unit. Ordinary mass scale. Open circles: observations or key values. Logarithmic coordinate scale. Open circles: observations or key values. q12=2⇒q=2121. Three steps give f=185q3=185241≈220.0033 hertz. This illustrates the logarithmic pitch premise only, not an external music-tuning project. Written-work review:
Use the one-kilogram reference to form three positive, dimensionless mass ratios and calculate their common-log coordinates. Supply the table pairing every original mass and its coordinate.
Show both an ordinary mass-scale plot and a log− coordinate plot with labeled axes and identifiable objects. Explain how the ordinary scale crowds small masses and the log scale represents multiplicative spacing; negative coordinates are not negative masses.
Use the quotient log law to derive the coordinate increment for multiplying mass by 15.84893192; explain that equal coordinate differences represent equal physical ratios.
Derive the common pitch factor by requiring twelve equal steps to double frequency, then apply three steps to 185 hertz. Give the final frequency in hertz and retain positive mass and frequency domains.
Numerical grading: accept equivalent exact expressions and final values within one unit in the last decimal displayed in the worked solution; for scientific notation apply that tolerance to the mantissa, and allow nearest-whole counts when requested. Preserve intermediate precision, stated units and interval endpoints. Assess each derivation and explanation separately; a selected summary alone is not a complete response.
Textbook reference: Stewart, Redlin & Watson, Precalculus: Mathematics for Calculus, 7th ed. (Cengage, 2016). The explanations and practice here are original, organized by textbook section.