NavGanitMateby NavGreen Compute, AustraliaBased on Australian Curriculum: Mathematics v9.0
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Logarithmic scales

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A craft designer in Melbourne uses a logarithmic scale to measure the perceived contrast in a textile pattern, defined by C = log10(R), where R is the ratio of light to dark intensity. If one design has a light-to-dark intensity ratio of 10^75 and another has a ratio of 10^52, what is the exact difference in perceived contrast between the two designs?

Express your answer as a single exact logarithmic expression.

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Equal spacing for equal multiplicative factors. On this base-ten logarithmic scale, 10, 100 and 1000 are equally spaced because each step multiplies the value by 10. Their logarithms are 1, 2 and 3.101001000×10×10log = 1log = 2log = 3
On this base-ten logarithmic scale, 10, 100 and 1000 are equally spaced because each step multiplies the value by 10. Their logarithms are 1, 2 and 3.

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Worked solution

  1. The perceived contrast C is defined by the logarithmic formula C = log10(R), where R is the light-to-dark intensity ratio.
  2. For the first design, R1 = 10^75. Substitute into the formula: C1 = log10(10^75).
  3. Apply the logarithmic identity log10(10^x) = x. Therefore, C1 = 75.
  4. For the second design, R2 = 10^52. Substitute into the formula: C2 = log10(10^52).
  5. Apply the same logarithmic identity: log10(10^52) = 52. Therefore, C2 = 52.
  6. The difference in perceived contrast is C1 - C2 = 75 - 52.
  7. Perform the subtraction: 75 - 52 = 23.
  8. The difference in perceived contrast is 23 units on the logarithmic scale.
  9. The question asks for the exact difference expressed as a single exact logarithmic expression.
  10. The difference is log10(10^75) - log10(10^52).
  11. Use the logarithmic identity log10(a) - log10(b) = log10(a/b). Therefore, the difference is log10(10^75 / 10^52).
  12. Simplify the exponent: 10^75 / 10^52 = 10^(75-52) = 10^23. Thus, the exact difference is log10(10^23), which evaluates to 23. The required single exact logarithmic expression is log10(10^23).

Reference answer: log10(10^23)

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