How do you compare two events on a logarithmic scale?

Subtract the two readings to get the difference d, then the ratio of the underlying quantities is ten to the power d for the Richter and pH scales, or ten to the power d divided by 10 for decibels. The difference in readings is never simply ten times d.

How do you invert a logarithmic scale to find the intensity?

Write the defining equation in index form. For M = log base 10 of (I divided by I0), a magnitude M gives I divided by I0 equal to ten to the power M. For L = 10 log base 10 of (I divided by I0), divide the level by 10 first, then I divided by I0 equals ten to the power L over 10.

How do you compare two events on a logarithmic scale?

Subtract the two readings to get the difference d, then the ratio of the underlying quantities is ten to the power d for the Richter and pH scales, or ten to the power d divided by 10 for decibels. The difference in readings is never simply ten times d.

How do you invert a logarithmic scale to find the intensity?

Write the defining equation in index form. For M = log base 10 of (I divided by I0), a magnitude M gives I divided by I0 equal to ten to the power M. For L = 10 log base 10 of (I divided by I0), divide the level by 10 first, then I divided by I0 equals ten to the power L over 10.

Year 12 Methods (Unit 3 & 4) Exponential and logarithmic functions

Applications of logarithmic functions

Q: What is a logarithmic scale?

A logarithmic scale measures a quantity by its logarithm, so that a quantity spanning many powers of ten is replaced by a small, readable number. Equal steps on the scale correspond to equal ratios of the quantity: on the Richter scale, decibel scale and pH scale a difference of one unit means the underlying quantity changes by a factor of ten.

Q: How does the Richter scale measure earthquakes?

An earthquake's magnitude is M = log base 10 of (I divided by I0), where I is its intensity and I0 is a fixed reference intensity. Because it is a base-ten logarithm, an increase of one magnitude unit is a tenfold increase in intensity, so a magnitude 7 earthquake is ten to the power three, or 1000, times as intense as a magnitude 4 earthquake.

Q: How is loudness measured in decibels?

The sound level is L = 10 log base 10 of (I divided by I0) decibels, where I0 is the threshold intensity of hearing. The factor of 10 means that a rise of 10 decibels corresponds to ten times the intensity, and a rise of 20 decibels to one hundred times the intensity.

Q: What does pH measure?

The pH of a solution is pH = negative log base 10 of the hydrogen-ion concentration in moles per litre. Because of the minus sign, a lower pH is more acidic. Pure water has pH 7; each decrease of one pH unit multiplies the hydrogen-ion concentration by ten.

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In Year 12 Mathematical Methods (Queensland, QCAA), several real-world scales are logarithmic: the Richter scale \(M=\log_{10}\!\left(\dfrac{I}{I_0}\right)\), the decibel scale \(L=10\log_{10}\!\left(\dfrac{I}{I_0}\right)\), and \(\text{pH}=-\log_{10}[\text{H}^+]\). On each, a difference of \(1\) unit is a factor of \(10\) in the underlying quantity — so these applications compute a reading from a ratio, compare two events by orders of magnitude, and invert a logarithm to recover an intensity or concentration.

A logarithmic scale measures a quantity by its logarithm. Quantities such as earthquake intensity, sound intensity and acidity range over many powers of \(10\), so replacing the quantity with \(\log_{10}\) of it gives a small, readable number in which equal steps are equal ratios.

The Richter scale gives an earthquake magnitude \(M=\log_{10}\!\left(\dfrac{I}{I_0}\right)\), where \(I\) is intensity and \(I_0\) a reference intensity. The decibel scale gives loudness \(L=10\log_{10}\!\left(\dfrac{I}{I_0}\right)\) — note the factor \(10\), where \(I_0\) is the threshold of hearing. The pH scale gives \(\text{pH}=-\log_{10}[\text{H}^+]\), with \([\text{H}^+]\) the hydrogen-ion concentration in moles per litre; the minus sign makes a lower pH more acidic, and pure water is \(\text{pH}=7\).

Because these are base-\(10\) logarithms, a difference of \(1\) unit corresponds to a factor of \(10\). To compare two events, subtract the readings and raise \(10\) to the difference; to invert, rewrite the defining equation in index form.

Key idea. On a base-\(10\) log scale a difference of \(1\) unit is a factor of \(10\): \(M=\log_{10}(I/I_0)\), \(L=10\log_{10}(I/I_0)\) dB, \(\text{pH}=-\log_{10}[\text{H}^+]\).

\(M=\log_{10}(I/I_0)\): increasing, through \((1,0)\), defined for \(I/I_0>0\)

\(\text{pH}=-\log_{10}[\text{H}^+]\): decreasing, so lower pH means larger \([\text{H}^+]\)

Earthquake magnitude (Richter scale), intensity \(I\), reference \(I_0\):

\[M=\log_{10}\!\left(\dfrac{I}{I_0}\right) \qquad\Longleftrightarrow\qquad \dfrac{I}{I_0}=10^{M}\]

M = \log_{10} (I / I_{0})

Sound level in decibels (note the factor \(10\)):

\[L=10\log_{10}\!\left(\dfrac{I}{I_0}\right) \qquad\Longleftrightarrow\qquad \dfrac{I}{I_0}=10^{L/10}\]

L = 10 \log_{10} (I / I_{0})

Acidity (pH), concentration \([\text{H}^+]\) in moles/litre:

\[\text{pH}=-\log_{10}[\text{H}^+] \qquad\Longleftrightarrow\qquad [\text{H}^+]=10^{-\text{pH}}\]

pH = - \log_{10} [H^{+}]

Comparing two events. Difference in readings \(d\) gives a ratio \(10^{d}\) (Richter, pH) or \(10^{d/10}\) (decibels). Example: two magnitudes differing by \(3\) differ in intensity by \(10^{3}=1000\).

How to compare two events on a log scale

Subtract the readings. Find the difference \(d\) of the two magnitudes / levels / pH values.
Raise \(10\) to the difference. The ratio of the underlying quantities is \(10^{d}\) for Richter and pH, or \(10^{d/10}\) for decibels (the factor \(10\) is undone first).
Evaluate with technology. For a whole-number difference the answer is an exact power of \(10\); otherwise use a calculator, e.g. \(10^{0.6}\approx 3.98\).

Inverting a log scale. To recover the quantity from a reading, write the defining equation in index form: \(M=\log_{10}(I/I_0)\Rightarrow I/I_0=10^{M}\); for decibels divide by \(10\) first, \(I/I_0=10^{L/10}\); for pH, \([\text{H}^+]=10^{-\text{pH}}\).

Example 1 — pH from concentration

A solution has \([\text{H}^+]=1\times10^{-5}\) moles/litre. Find its pH and state whether it is acidic.

Solution

Substitute and use \(\log_{10}10^{n}=n\).

\(\text{pH}\)	\(=\)	\(-\log_{10}(10^{-5})\)
	\(=\)	\(-(-5)=5\)

Since \(5<7\), the solution is acidic.

pH = 5

Example 2 — Comparing earthquakes

On the Richter scale, how many times as intense is a magnitude \(7.5\) earthquake as a magnitude \(5.5\) one?

Solution

The ratio is \(10\) to the difference in magnitude.

\(\dfrac{I_1}{I_2}\)	\(=\)	\(10^{\,7.5-5.5}\)
	\(=\)	\(10^{2}=100\)

It is \(100\) times as intense.

10^{2} = 100

Example 3 — Decibel level

A sound has intensity \(I=10^{8}\,I_0\). Find its level in decibels, using \(L=10\log_{10}(I/I_0)\).

Solution

Substitute the ratio and keep the factor \(10\).

\(L\)	\(=\)	\(10\log_{10}(10^{8})\)
	\(=\)	\(10\times 8=80\)

The level is \(80\) dB.

L = 80

Example 4 — Inverting a log scale

An earthquake measures \(M=6\). Express its intensity as a multiple of \(I_0\).

Solution

Write \(M=\log_{10}(I/I_0)\) in index form.

\(6\)	\(=\)	\(\log_{10}(I/I_0)\)
\(I/I_0\)	\(=\)	\(10^{6}\)

The intensity is \(10^{6}\,I_0\).

I = 10^{6} I_{0}

Common pitfalls

A difference of \(1\) unit is a factor of \(10\), not a difference of \(10\). A magnitude \(6\) earthquake is \(10^{2}=100\) times as intense as a magnitude \(4\), not \(1.5\) times or \(20\) times.

Decibels carry the factor \(10\). For \(L=10\log_{10}(I/I_0)\), a \(20\) dB rise is \(10^{20/10}=100\) times the intensity. Divide the level by \(10\) before raising \(10\) to it.

pH runs backwards. Because \(\text{pH}=-\log_{10}[\text{H}^+]\), a smaller pH is more acidic; each drop of \(1\) pH unit multiplies \([\text{H}^+]\) by \(10\).

Frequently asked questions

What is a logarithmic scale?

A scale that measures a quantity by its \(\log_{10}\), so equal steps are equal ratios. A difference of \(1\) unit means the quantity changes by a factor of \(10\).

How does the Richter scale measure earthquakes?

\(M=\log_{10}(I/I_0)\). A one-unit increase is a tenfold increase in intensity, so magnitude \(7\) is \(10^{3}=1000\) times magnitude \(4\).

How is loudness measured in decibels?

\(L=10\log_{10}(I/I_0)\) dB. The factor \(10\) means a \(10\) dB rise is \(\times 10\) intensity, and \(20\) dB is \(\times 100\).

What does pH measure?

\(\text{pH}=-\log_{10}[\text{H}^+]\), the acidity of a solution. Lower pH is more acidic; pure water is \(\text{pH}=7\).

How do you compare two events on a log scale?

Subtract the readings to get \(d\); the ratio is \(10^{d}\) (Richter, pH) or \(10^{d/10}\) (decibels).

How do you invert a log scale to find the intensity?

Write it in index form: \(I/I_0=10^{M}\) (Richter), \(I/I_0=10^{L/10}\) (decibels), \([\text{H}^+]=10^{-\text{pH}}\) (pH).

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