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Use a formula-bearing logarithmic scale

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To find a raw quantity from a level, what should happen first?

\[L=c\log_{10}\left(\frac{Q}{Q_0}\right),\quad c\ne0,\quad Q,Q_0>0\]
Year 12, VCE Units 3 & 4, Chapter 4, 4 difficulty levels, Calculator allowedBuild a worksheet

Formula-bearing logarithmic scales

A logarithmic scale compares a positive raw quantity Q with a positive reference quantity Q₀. The scale coefficient c is nonzero. Substitute for a forward calculation, or isolate the logarithm and use a power of 10 when reversing the formula

\[L=c\log_{10}\left(\frac{Q}{Q_0}\right),\quad c\ne0,\quad Q,Q_0>0\]
  1. Substitute the known quantity Use the ratio of the raw quantity to the reference quantity in the stated formula \(\displaystyle L=2\log_{10}\left(\frac{1000}{100}\right)=2\log_{10}(10)=2\)
  2. Divide by the coefficient When finding a raw quantity from a level, divide the level by c before using the inverse logarithm \(\displaystyle 6=2\log_{10}\left(\frac{Q}{100}\right)\quad\Rightarrow\quad \log_{10}\left(\frac{Q}{100}\right)=3\)
  3. Use the base-ten inverse Replace a logarithm equal to r with a power of 10 and then multiply by the reference quantity \(\displaystyle \log_{10}\left(\frac{Q}{Q_0}\right)=r\quad\Rightarrow\quad Q=Q_0\times10^r\)
  4. Keep a negative coefficient A leading negative coefficient reverses the direction between the level and the raw quantity \(\displaystyle L=-\log_{10}\left(\frac{Q}{Q_0}\right)\quad\Rightarrow\quad Q_2>Q_1\Longrightarrow L_2<L_1\)

Worked example

Find the raw quantity for the stated level

\[L=3\log_{10}\left(\frac{Q}{10}\right),\quad L=6\]
  1. \(\displaystyle 6=3\log_{10}\left(\frac{Q}{10}\right)\)Substitute the stated level into the scale formula
  2. \(\displaystyle \log_{10}\left(\frac{Q}{10}\right)=\frac{6}{3}=2\)Divide by the scale coefficient to isolate the logarithm
  3. \(\displaystyle \frac{Q}{10}=10^2=100\)Use the inverse of the base-ten logarithm
  4. \(\displaystyle Q=10\times100=1000\)Multiply the ratio by the reference quantity to obtain the raw quantity

Common mistakes

Keeping the coefficient as a factor
✕\(\displaystyle 6=2\log_{10}\left(\frac{Q}{100}\right)\quad\Rightarrow\quad Q=100\times10^6=100000000\)✓\(\displaystyle 6=2\log_{10}\left(\frac{Q}{100}\right)\quad\Rightarrow\quad \log_{10}\left(\frac{Q}{100}\right)=3\quad\Rightarrow\quad Q=100\times10^3=100000\)
The scale coefficient must be divided out before the logarithm is changed to a power of 10
Reading a level difference as a factor
✕\(\displaystyle L_A-L_B=2\quad\Rightarrow\quad \frac{Q_A}{Q_B}=2\)✓\(\displaystyle L_A-L_B=2\quad\Rightarrow\quad \frac{Q_A}{Q_B}=10^{\frac{2}{c}}\)
A level difference becomes a factor only after division by c and exponentiation with base 10
Dropping a leading negative
✕\(\displaystyle L=-\log_{10}\left(\frac{Q}{Q_0}\right),\quad Q_2>Q_1\quad\Rightarrow\quad L_2>L_1\)✓\(\displaystyle L=-\log_{10}\left(\frac{Q}{Q_0}\right),\quad Q_2>Q_1\quad\Rightarrow\quad L_2<L_1\)
The negative coefficient reverses the level direction as the raw quantity increases

Difficulty levels

introUses an exact power of 10 with an integer level in one direction
standardFinds a level or raw quantity using a non-unit coefficient and stated decimal precision
challengeCompares two raw quantities by adjusting a level difference for the scale coefficient
examCombines an inversion, a comparison and a classification of the larger raw quantity

Questions about this topic

Why is Q divided by Q₀ in a logarithmic scale?+

The ratio compares the raw quantity with the reference quantity. When the quantities are equal, their ratio is 1 and its base-ten logarithm is 0

How does a level difference give a factor?+

Subtract the levels, divide by the scale coefficient c, and use 10 raised to that result

What does a negative scale coefficient do?+

It reverses the direction: increasing the raw quantity makes the logarithmic level decrease

Can I use a calculator?+

A calculator is allowed unless a question is marked no calculator

Which year level and chapter is this?+

Year 12, VCE Units 3 & 4, Mathematical Methods, chapter 4: Logarithmic functions

In this chapter

Connect exponential and logarithmic formsApply logarithm laws and inverse identitiesEvaluate logarithms and antilogarithmsExpress one logarithmic quantity in terms of anotherUse a formula-bearing logarithmic scaleSolve exponential equations and inequalities exactlySolve exponential and logarithmic equationsSolve literal logarithmic and indicial equationsAnalyse exponential and logarithmic graphsConstruct exponential and logarithmic inverse pairsInterpret and use an exponential modelInterpret and use a logarithmic modelDetermine and use a logarithmic model