Exponential functions and logarithms: Logarithms
Applications of ln
Like its counterpart, the exponential function, you encounter the natural logarithm in many applications of mathematics in life sciences, if only because you want to solve exponential equations. Exponential equations are often transformed by making use of the natural logarithm into linear relationships between related quantities. As an example we mention the Arrhenius equation.
Arrhenius equation The Arrhenius equation \[k=A e^{\frac{E_a}{RT}}\] can be rewritten in terms of the natural logarithm as \[\ln(k)=\frac{E_a}{R}\cdot \frac{1}{T}+\ln(A)\] What one has obtained by this is a linear relationship between the natural logarithm of the rate constant \(k\) and the reciprocal value of the temperature \(T\), in which the parameters depend on the activation energy \(E_a\), the molar gas constant \(R\) and the constant \(A\).
A second example of application of the natural logarithm comes from electrochemistry, but we will discuss this in the context of cell biology. In the third section of this chapter about 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.
The Nernst equation can also be written differently on the basis of calculation rules for the natural logarithm. From \(\ln(1/x)=-\ln(x)\) follows: \[E_\mathrm{ion}=-\frac{RT}{zF}\cdot \ln\left(\frac{[C]_i}{[C]_e}\right)\]
The third context is psychophysics.
Weber-Fechner law The Weber-Fechner law is a general rule for the relationship between the strength of physical impulses on the human senses (stimuli) and the intensity of the accompanying sensations. The law prescribe that when physical impulses increase with constant proportions, then the sensations increase with constant differences.
In mathematical language: when a stimulus \(S\) varies according to a geometric sequence \[S=1,a,a^2, a^3, \ldots\] hen perception \(P\) hereof change as an arithmetic sequence \[P=p, p+b, p+2b, p+3b, \ldots\] In yet more formal language, the relationship can be written as follows, in terms of logarithms: \[P=k\cdot\ln\left(\frac{S}{S_0}\right)\] where \(S_0\) is the threshold of the stimulus below which no sensation takes place.
