Open MathNSWGitHub ↗

Logarithmic scales

Recognise when a logarithmic scale is appropriate for large-range data; solve problems using the decibel scale for sound intensity, the Richter scale for earthquakes, the pH scale for acidity, and stellar magnitude for star brightness.

Worked examples

Decibel calculations

Straightforward

Problem

A busy restaurant has a background noise intensity of 10−510^{-5} W/m². A quiet library has intensity 10−910^{-9} W/m². Calculate the loudness of each in decibels, and find the difference in loudness. Use L=10log⁡10 ⁣(II0)L = 10\log_{10}\!\left(\dfrac{I}{I_0}\right) with I0=10−12I_0 = 10^{-12} W/m².

Richter scale comparison

Moderate

Problem

An earthquake of magnitude 5.05.0 causes minor damage. How much stronger (in terms of wave amplitude) would a magnitude 7.57.5 earthquake be? Round to 1 decimal place.

pH and hydrogen ion concentration

Challenging

Problem

A swimming pool manager measures pH 7.87.8 (slightly alkaline). Adding a chemical lowers the pH to 6.86.8. By what factor does the [H+][\text{H}^+] concentration increase? Round to 1 decimal place.

Practise

Q1·Straightforward
The loudness LL (in decibels) of a sound is given by L=10log⁡10 ⁣(II0)L = 10\log_{10}\!\left(\dfrac{I}{I_0}\right), where II is the sound intensity in W/m² and I0=10−12I_0 = 10^{-12} W/m² is the threshold of hearing.

A sound has intensity I=10−8I = 10^{-8} W/m². Find its loudness in decibels.
Q2·Straightforward
A jet engine produces a sound intensity of 100100 W/m². Using L=10log⁡10 ⁣(II0)L = 10\log_{10}\!\left(\dfrac{I}{I_0}\right) with I0=10−12I_0 = 10^{-12} W/m², find the loudness in decibels.
Q3·Straightforward
The Richter scale magnitude of an earthquake is M=log⁡10 ⁣(AA0)M = \log_{10}\!\left(\dfrac{A}{A_0}\right), where AA is the maximum wave amplitude and A0A_0 is a reference amplitude.

Two earthquakes have magnitudes 66 and 88 respectively. How many times greater is the wave amplitude of the larger earthquake compared with the smaller one?
Q4·Straightforward
The pH of a solution is defined as pH=−log⁡10[H+]\text{pH} = -\log_{10}[\text{H}^+], where [H+][\text{H}^+] is the hydrogen ion concentration in mol/L.

Vinegar has pH 33 and drinking water has pH 77. How many times greater is the [H+][\text{H}^+] concentration in vinegar than in water?
Q5·Moderate
A sound has loudness L=65L = 65 dB. Using L=10log⁡10 ⁣(II0)L = 10\log_{10}\!\left(\dfrac{I}{I_0}\right) with I0=10−12I_0 = 10^{-12} W/m², find the intensity II.

If I=10nI = 10^n W/m², state the value of nn.
Q6·Moderate
An earthquake has magnitude 7.57.5 on the Richter scale. A second earthquake produces wave amplitudes 100100 times greater. What is the magnitude of the second earthquake?
Q7·Moderate
Two identical machines each produce a sound of 7070 dB. When both run simultaneously the intensities add. What is the combined loudness in dB, rounded to 1 decimal place?

(Use log⁡102≈0.3010\log_{10} 2 \approx 0.3010.)
Q8·Moderate
The apparent magnitude scale for stars is defined so that a difference of 55 magnitudes corresponds to a brightness ratio of exactly 100100. A more precise formula is:
m1−m2=−2.5log⁡10 ⁣(F1F2)m_1 - m_2 = -2.5\log_{10}\!\left(\frac{F_1}{F_2}\right)

where FF denotes flux (brightness) and smaller magnitude means brighter.

Star A has apparent magnitude 2.02.0 and star B has apparent magnitude 4.54.5. How many times brighter is star A than star B?
Q9·Moderate
A solution has hydrogen ion concentration [H+]=4.5×10−4[\text{H}^+] = 4.5 \times 10^{-4} mol/L. Find the pH of the solution, rounded to 2 decimal places.

(Use log⁡104.5≈0.6532\log_{10} 4.5 \approx 0.6532.)
Q10·Challenging
A noise-reduction regulation requires a factory to lower its sound level by 1515 dB. By what factor must the sound intensity be reduced?

Round your answer to 2 decimal places.
Q11·Challenging
An earthquake in region P has magnitude 6.26.2. An earthquake in region Q has magnitude 4.74.7. How many times greater is the wave amplitude of the earthquake in P than in Q?

Round your answer to 2 decimal places.
Q12·Challenging
Star X is 55 times brighter (greater flux) than star Y. Star Y has apparent magnitude 3.03.0. Find the apparent magnitude of star X, rounded to 2 decimal places.

Use the formula m1−m2=−2.5log⁡10(F1/F2)m_1 - m_2 = -2.5\log_{10}(F_1/F_2) and log⁡105≈0.6990\log_{10} 5 \approx 0.6990.