4.7 Logarithmic Scales
Ideas in this lesson
- Interpret logarithmic coordinates and equal ratios
- Convert pH, compare acidity and transform intervals
- Convert earthquake amplitudes and magnitude differences
- Convert decibels and compare physical intensities
- Prove and apply the inverse-square sound relation
Interpret logarithmic coordinates and equal ratios · Worked example
A custom mass coordinate is defined by \(L=\log_{10}(\dfrac{W}{W_0})\), where \(W_0=0.25\,\mathrm{kg}\). Recover the mass whose coordinate is \(2.6\). If all masses are recorded in grams instead, show why the coordinate stays unchanged when the reference is converted too, but the logarithm of the numerical mass alone shifts. Find the coordinate change for multiplying mass by 5. In a 12-steps-per-doubling frequency scale, how many steps separate 216 Hz and 367.2 Hz? State whether a fractional step count is allowed by the mathematical scale. 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.
Answer: Mass 99.5268 kg; coordinates unchanged under consistent unit conversion, while the raw numerical-mass logarithm shifts by 3. Log change 0.69897; pitch separation 9.1864 steps.
\(10^{L}=\dfrac{W}{W_0}\Rightarrow W=W_0\,10^{L}=0.25\,10^{2.6}\approx 99.5268\) kg.
\(\log_{10}\left(\dfrac{1000W}{1000W_0}\right)=\log_{10}(\dfrac{W}{W_0})\). The dimensionless ratio does not change. By contrast, \(\log_{10}(1000w)=3+\log_{10}w\) for a numerical mass value \(w\) in kilograms.
\(\Delta L=\log_{10}(5)\approx 0.69897\). Equal coordinate differences measure multiplicative ratios.
\(\dfrac{f_2}{f_1}=2^{\dfrac{s}{12}}\Rightarrow s=\dfrac{12\ln(\dfrac{f_2}{f_1})}{\ln2}\approx 9.1864\) steps. Fractional steps are allowed on the continuous mathematical scale, even if a discrete keyboard contains only selected pitches. All masses, references and frequencies must be positive.
Written-work review:
\(\log_{10}\left(\dfrac{1000W}{1000W_0}\right)=\log_{10}(\dfrac{W}{W_0})\). The dimensionless ratio does not change. By contrast, \(\log_{10}(1000w)=3+\log_{10}w\) for a numerical mass value \(w\) in kilograms.
\(\Delta L=\log_{10}(5)\approx 0.69897\). Equal coordinate differences measure multiplicative ratios.
\(\dfrac{f_2}{f_1}=2^{\dfrac{s}{12}}\Rightarrow s=\dfrac{12\ln(\dfrac{f_2}{f_1})}{\ln2}\approx 9.1864\) steps. Fractional steps are allowed on the continuous mathematical scale, even if a discrete keyboard contains only selected pitches. All masses, references and frequencies must be positive.
Written-work review:
- Invert the common-log coordinate 2.6 by exponentiating base ten and multiplying by the 0.25-kilogram reference; report a mass, not merely a dimensionless power.
- Convert both mass and reference from kilograms to grams and show cancellation of the common factor in their ratio. Separately show why the logarithm of the numerical mass alone increases by three; do not conflate it with the dimensionless coordinate.
- Derive the coordinate increment for multiplying mass by 5, and solve the twelve-steps-per-doubling equation for the separation between 216 and 367.2 hertz.
- Explain that the mathematical pitch scale permits fractional step counts, even though a particular discrete instrument might not supply them. Require positive masses, reference and frequencies.
- 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.