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

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A school in Brisbane orders 56 science kits and 67 mathematics workbooks. The cost of each science kit is $10^(log10(67) - 2) and each workbook is $10^(log10(56) - 3).

What is the exact total cost of the order? Express your answer as a single exact value using logarithmic and exponential properties.

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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. Given: 56 science kits, 67 mathematics workbooks. Cost per science kit = $10^(log10(67) - 2), cost per workbook = $10^(log10(56) - 3). Need to find total cost as a single exact value using logarithmic and exponential properties.
  2. Use the logarithmic identity: 10^(log10(a) - b) = a / 10^b. Apply to science kit cost: 10^(log10(67) - 2) = 67 / 10^2 = 67 / 100 = 0.67.
  3. Apply the same identity to workbook cost: 10^(log10(56) - 3) = 56 / 10^3 = 56 / 1000 = 0.056.
  4. Calculate total cost for science kits: 56 × 0.67 = 37.52.
  5. Calculate total cost for workbooks: 67 × 0.056 = 3.752.
  6. Add both costs: 37.52 + 3.752 = 41.272.
  7. Express 41.272 as a fraction. 41.272 = 41272 / 1000.
  8. Simplify 41272 / 1000 by dividing numerator and denominator by 8: 41272 ÷ 8 = 5159, 1000 ÷ 8 = 125. So 5159 / 125.
  9. Check if 5159 and 125 have common factors. 125 = 5^3. 5159 ÷ 5 = 1031.8 → not divisible. So 5159/125 is in simplest form.
  10. Verify using logarithmic-exponential identity directly: total cost = 56 × 10^(log10(67) - 2) + 67 × 10^(log10(56) - 3).
  11. Substitute: 56 × (67 / 100) + 67 × (56 / 1000) = (56×67)/100 + (67×56)/1000. Note 56×67 = 3752 in both terms.
  12. So total = 3752/100 + 3752/1000 = 37520/1000 + 3752/1000 = 41272/1000 = 5159/125. Final answer is 5159/125 dollars.

Reference answer: 5159/125

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