### Exponential functions and logarithms: Logarithms

### Applications of the logarithm with base 10

Like its counterpart, an exponential function, you encounter the logarithmic function in many applications of mathematics in life sciences, if only because you want to solve an exponential equation. It almost always boils down to logarithms with base 10. For the first two examples we choose a chemical contex: acidity and the Lambert-Beer law.

pH as measure of acidity A measure of acidity of an aqueous solution is the concentration \(\bigl[\mathrm{H}_3\mathrm{O}^{+}\bigr]\) in mol/L, but this is not really convenient because you are then always working with small values. For this reason, the **pH** has been introduced as a measure for acidity: \[\mathrm{pH} = -\log_{10}\biggl(\bigl[\mathrm{H}_3\mathrm{O}^{+}\bigr]\biggr)\] If, for example, \(\bigl[\mathrm{H}_3\mathrm{O}^{+}\bigr]=10^{-7}\mathrm{mol/L}\), then the pH value is equal to \(\mathrm{pH}=-\log_{10}\bigl(10^{-7}\bigr)=7\cdot\log_{10}(10) = 7\) and there is a neutral aqueous solution. Acidic solutions having a pH less than 7, basic solutions has a pH greater than 7. The pH of a solution thus decreases by 1 if the concentration \(\bigl[\mathrm{H}_3\mathrm{O}^{+}\bigr]\) is multiplied by 10. The concentration \(\bigl[\mathrm{H}_3\mathrm{O}^{+}\bigr]\) can be calculated from the pH value: \[\bigl[\mathrm{H}_3\mathrm{O}^{+}\bigr]=10^{-\mathrm{pH}}\] In order to avoid working with negative exponents in scientific notation our acidity constant \(K_a\) and base constant \(K_b\) are replaced by their opposite logarithms and renamed: \[\mathrm{p}K_a = -\log_{10}(K_a), \quad \mathrm{p}K_a = - \log_{10}(K_a)\]

Lambert-Beer law A number of observations play a role in this example.

- a stronger solution of a coloured substance looks darker than a weaker solution of the same substance.
- a thin layer of a coloured solution is paler than a thick layer of the same solution.
- different substances have different colors.

This observations can be quantified in a spectrophotometer. Some chemical substances have the property that they absorb light of a particular wavelength in the spectrum. The concentration of such substance in a mixture can be determined by investigating the absorbance, also known as extinction, of monochromatic light of an appropriate wavelength. The extinction \(E\) is defined in the following relation as \[E=\log_{10}\left(\frac{I_0}{I}\right)\] where \(I_0\) is the original amount of light and \(I\) is the amount of light which is transmitted through the sample. The** Lambert-Beer law** states that the extinction is directly proportional to the concentration of the light-absorbing material: \[E=\varepsilon\cdot C\cdot l\] where \(\varepsilon\) is the molar extinction coefficient, \(C\) is the concentration of the substance, and \(l\) is the path length of the cuvette.

A familiar example of using logarithms comes from seismology: the moment magnitude scale.

Moment magnitude scale for earthquakes** The Richter scale** is a known measure of the strength of an earthquake. Seismologists, however, use the later defined **moment magnitude scale.** This scale represents the strength of an earthquake as a number \(n\) that can be calculated with the formula \[n=\frac{2}{3}\cdot\log_{10}(M)-6\] where \(M\) is the seismic moment, which is directly proportional to the radiated energy of the earthquake.

A major application of logarithms is the ratio of magnitudes on a log scale.

decibel as measure of sound intensity The noise level \(L\) is, for example, defined as the logarithmic ratio of the absolute value of the sound intensity \(J\) and a reference value \(J_0\) and expressed in **decibel** **(dB):** \[L=10\cdot \log_{10}\left(\frac{J}{J_0}\right)\]

The final example of applying logarithms is to rewrite the Nernst equation in the context of cell biology. In the third section of this chapter on bioelectricity we discuss this in more detail.

Nernst equation In a situation in which a cell membrane is only permeable to one type of ion, an electric potential difference will develop between the two sides of the membrane. This potential difference is in a kind of equilibrium (that is, equal and opposite) with the concentration gradient. The value of the potential depends on the ion concentrations on both sides of the membrane, of the charge per ion (valence) and of the temperature \(T\) in accordance with the **Nernst equation** for the calculation of the equilibrium potential \[E_\mathrm{ion}=\frac{RT}{zF}\cdot \ln\left(\frac{[C]_e}{[C]_i}\right)\] where \(R\) is the molar gas constant, \(T\) is the absolute temperature, \(z\) is the valence of the ion, \(F\) is the Faraday constant, and \([C]_e\) and \([C]_i\) are the extracellular and intracellular concentration, respectively. In living cells, the potential on the inside of the cell is negative relative to the potential on the outside for cations. The difference in potential in a nerve cell of a squid is at a temperature of 37 degrees Celsius for the potassium ion approximately 75 mV.

This equation you regularly encounter with \(\log_{10}\) instead of the natural logarithm \(\ln\) ; the Nernst equation is then

\[\begin{aligned}E_\mathrm{ion}& =\frac{RT}{zF}\cdot \ln\left(\frac{[C]_e}{[C]_i}\right)\\ \\ &= \ln(10)\cdot \frac{RT}{zF}\cdot \log_{10}\left(\frac{[C]_e}{[C]_i}\right) \\ \\ &\approx 2.303 \cdot \frac{RT}{zF}\cdot \log_{10}\left(\frac{[C]_e}{[C]_i}\right) \\ \\ &\approx 59.1 \cdot \log_{10}\left(\frac{[C]_e}{[C]_i}\right)\quad\text{(in mV) for a monovalent cation at 298 K}\end{aligned}\]