Lesson 2: Logarithmic Scales

New Concept

Logarithmic scales are powerful tools for visualizing data across vast value ranges. By plotting the logarithm of a quantity, we can easily compare values that differ by many orders of magnitude on a single, manageable axis.

What’s next

You’ll soon practice plotting values and interpreting log scales through interactive examples involving pH, decibels, and earthquake magnitudes.

Property

To get around the problem of plotting values with a very wide range, we can plot the log of the mass, instead of the mass itself. The scale is called a logarithmic scale, or log scale. The tick marks are labeled with powers of 10. When we plot the logarithm of a value, we are really plotting the power of 10 that gives its mass. For example, for a value $x$, its position on the log scale is at $\log x$.

Examples

• To plot the number 8000 on a base-10 log scale, we calculate $\log 8000 \approx 3.9$. We then place a mark at position 3.9 on the scale, between the major tick marks for $10^3$ and $10^4$.
• A value of 0.05 is plotted by finding its logarithm, $\log 0.05 \approx -1.3$. This point would be placed on the scale between the tick marks for $10^{-2}$ and $10^{-1}$.
• To compare the masses of a 0.2 kg squirrel and a 40,000 kg whale, we plot their logs. $\log 0.2 \approx -0.7$ and $\log 40000 \approx 4.6$. These two points fit easily on a scale from -1 to 5.

Explanation

Log scales help us see everything at once, from tiny fractions to giant numbers. By plotting the exponent (the logarithm), we can fit an enormous range of values onto a single, manageable number line, making comparisons easy.

Property

Equal increments on a log scale do not correspond to equal differences in value, as they do on a linear scale. As we move from left to right on this scale, we multiply the value at the previous tick mark by 10. Moving up by equal increments on a log scale does not add equal amounts to the values plotted; it multiplies the values by equal factors.

Examples

• The physical distance on a log scale between 10 and 100 is the same as the distance between 100 and 1000. Both represent a multiplication by a factor of 10.
• The halfway point on a log scale between $10^4$ (10,000) and $10^5$ (100,000) is not 55,000. It represents the value $10^{4.5}$, which is approximately 31,623.
• On a log scale, the tick mark for 30 is much closer to 10 than the tick mark for 80 is to 100. This is because the values are spaced based on their ratios, not their differences.

Explanation

Unlike a regular ruler, a log scale is a 'multiplier' scale. Each major tick mark is 10 times bigger than the one before it. This is why integer labels get closer and closer as they approach the next power of 10.

Property

The pH scale is used by chemists to measure the acidity of a substance. The pH value is defined by the formula:

Examples

• If a cleaning solution has a hydrogen ion concentration of $[H^+] = 10^{-12}$, its pH is calculated as $\text{pH} = -\log_{10}(10^{-12}) = -(-12) = 12$.
• Tomato juice has a pH of about 4. This is 1000 times more acidic than pure water (pH 7), because the difference in pH is $7-4=3$, which corresponds to a factor of $10^3$.
• To find the hydrogen ion concentration of a liquid with a pH of 3.5, we solve $3.5 = -\log_{10}[H^+]$. This gives $[H^+] = 10^{-3.5} \approx 3.16 \times 10^{-4}$.

Explanation

The pH scale simplifies acidity by converting complex hydrogen ion concentrations into simple numbers, usually from 0 to 14. A lower pH means higher acidity, where each whole number step down represents a tenfold increase in acidity.

Property

The decibel scale, used to measure the loudness or intensity of a sound, is another example of a logarithmic scale. The perceived loudness of a sound is measured in decibels, $D$, by

Examples

• A quiet room has a sound intensity of $I = 10^{-10}$ watts/m². Its decibel level is $D = 10 \log_{10} \left(\frac{10^{-10}}{10^{-12}}\right) = 10 \log_{10}(10^2) = 10(2) = 20$ dB.
• A rock concert at 110 dB is compared to a vacuum cleaner at 70 dB. The difference of 40 dB corresponds to a factor of $10^{40/10} = 10^4$, meaning the concert is 10,000 times more intense.
• A sound is measured at 95 decibels. To find its intensity $I$, we solve $95 = 10 \log_{10} \left(\frac{I}{10^{-12}}\right)$. This gives $9.5 = \log_{10} \left(\frac{I}{10^{-12}}\right)$, so $I = 10^{9.5} \times 10^{-12} = 10^{-2.5}$ watts/m².

Explanation

The decibel scale measures sound intensity logarithmically, making it easier to handle the vast range from a faint whisper to a loud jet engine. An increase of 10 decibels means the sound is 10 times more intense.

Property

One method for measuring the magnitude of an earthquake compares the amplitude $A$ of its seismographic trace with the amplitude $A_0$ of the smallest detectable earthquake. The log of their ratio is the Richter magnitude, $M$. Thus,

Examples

• An earthquake measures 6.0 on the Richter scale and an aftershock measures 4.0. The main quake's amplitude was $10^{6-4} = 10^2 = 100$ times larger than the aftershock's amplitude.
• If an earthquake has a seismograph amplitude 800,000 times greater than the baseline $A_0$, its magnitude is $M = \log_{10} \left(\frac{800000 A_0}{A_0}\right) = \log_{10}(8 \times 10^5) \approx 5.9$.
• To compare an earthquake of magnitude 7.8 to one of magnitude 6.2, we find the ratio of their amplitudes: $10^{7.8-6.2} = 10^{1.6} \approx 39.8$. The first quake is about 40 times more powerful in amplitude.

Explanation

The Richter scale measures earthquake power on a logarithmic scale. Each whole number increase on the scale represents a tenfold increase in the measured amplitude of the seismic waves, making immense power differences easy to compare.