Logarithmic Scales

The quietest sound a person can hear and the sound of a jet engine differ in intensity by a factor of about $10^{12}$, a million million. If you drew that on ordinary graph paper with the quietest sound $1$ mm from the origin, how far away would the jet engine be? Give your answer in kilometres, and then say what you would do about it.

On an ordinary scale equal steps mean equal differences: $10, 20, 30$. On a logarithmic scale equal steps mean equal RATIOS: $10, 100, 1000$. Because a logarithm converts multiplication into addition, a quantity that multiplies by a fixed factor moves a fixed distance along the axis.

Audible sound intensity runs from about $10^{-12}$ watts per square metre at the threshold of hearing to about $1$ at the threshold of pain. That is a range of $10^{12}$ to one.

On a linear axis at $1$ mm for the quietest sound, the loudest would sit $10^{12}$ mm away, which is a million kilometres. No graph, ruler or dial can carry that.

Taking logarithms turns the range $10^{-12}$ to $1$ into the range $-12$ to $0$. Multiply by $10$ and shift, and you get the familiar decibel scale from $0$ to $120$: a number you can print on a dial.

Because $\log_{10} A - \log_{10} B = \log_{10}\left(\dfrac{A}{B}\right)$, subtracting two readings gives the logarithm of the ratio. Undo that logarithm and you have the ratio itself.

$$\frac{A}{B} = 10^{\,d} \quad \text{where } d \text{ is the difference in readings}$$

Two earthquakes measuring $7.0$ and $5.0$ differ by $d = 2$, so the ground moves $10^{2} = 100$ times as far in the larger one. A lemon at pH $2$ and black coffee at pH $5$ differ by $3$, so the lemon has $10^{3} = 1000$ times the hydrogen ion concentration.

Notice this is just the quotient law from Lesson 3, doing exactly what it was derived to do.

pH. $\text{pH} = -\log_{10}[\text{H}^{+}]$. The minus sign is there because the concentrations are tiny, so their logarithms are negative and the sign makes the readings positive. Because of it, a LOWER pH means a HIGHER concentration.

Richter magnitude. $+1$ in magnitude means ten times the amplitude of ground movement. Released energy grows faster still, by a factor of about $31.6$, which is $10^{1.5}$, for each whole step.

Decibels. $L = 10\log_{10}\left(\dfrac{I}{I_0}\right)$, with $I_0 = 10^{-12}$ W/m². The factor of $10$ in front means that $+10$ dB, not $+1$ dB, corresponds to ten times the intensity.

From quantity to reading. A solution has $[\text{H}^{+}] = 10^{-4}$ mol/L. Then $\text{pH} = -\log_{10}(10^{-4}) = -(-4) = 4$.

From reading to quantity. A solution has pH $9$. Then $-\log_{10}[\text{H}^{+}] = 9$, so $\log_{10}[\text{H}^{+}] = -9$ and $[\text{H}^{+}] = 10^{-9}$ mol/L.

Comparing two. Milk at pH $6.5$ and lemon juice at pH $2.5$ differ by $4$, so the lemon juice has $10^{4} = 10\,000$ times the hydrogen ion concentration.

Every one of these is Lesson 5's method: to undo a logarithm, raise ten to both sides.

Four quick problems. Work each one, then reveal the answer.

1. 1 Two earthquakes measure $6.0$ and $4.0$. How many times greater is the amplitude of the larger?

   Show answer
   Difference $2$, so the ratio is $10^{2}$.$100$ times

2. 2 A solution has $[\text{H}^{+}] = 10^{-3}$ mol/L. Find its pH.

   Show answer
   $\text{pH} = -\log_{10}(10^{-3}) = 3$.pH $3$

3. 3 Sound A is $1000$ times as intense as sound B. What is the difference in decibels?

   Show answer
   $10\log_{10}(1000) = 10 \times 3$.$30$ dB

4. 4 Solution X has pH $4$ and solution Y has pH $7$. Which is more acidic, and by what factor in $[\text{H}^{+}]$?

   Show answer
   Lower pH means more acidic; the difference of $3$ gives $10^{3}$.X, by $1000$ times