4.3. Logarithm Algebra

Here is a quick reminder of some logarithm algebra to be used with thermodynamics and kinetics. Recall that the logarithm function is the inverse operation of exponentiation. That is, the logarithm base \(b\) of \(b^x\) simply retuns the value \(x\)

\[\log_b(y) = \log_b(b^x) = x\]

As an example, for the base 10, we can see that \(10^{3.4} \approx 2511.9\) so that

\[\log_{10}(2511.9) \approx \log_{10}(10^{3.4}) = 3.4\]

We will frequently use natural logarithms. These are logarithms that use the transcendental number \(e \approx 2.718\) as the base and the function is symbolized as \(\ln\). The natural logarithm comes up frequently in systems with exponential growth.

Let’s consider the natural log of the base \(e\) raised to a power. We can symbolize an exponentiated nubmer using \(\exp(x)\) or \(e^x\). The natural log of these returns the value \(x\).

\[\ln(\exp(x)) = \ln(e^x) = x\]

Several useful common logarithm algebra properties using natural logarithms are shown in Table 4.6, below. You can derive these relationships simply by considering that logarithms are inverse exponentiation but they are stated here without derivation. They also work for other bases.

Table 4.6 Logarithm Algebra Properties

Property	Equation
logarithm addition	\(\ln(x) + \ln(y) = \ln(xy)\)
logarithm subtraction	\(\ln(x) - \ln(y) = \ln\left(\dfrac{x}{y}\right)\)
multiplication by a constant	\(a\ln(x) = \ln(x^a)\)
change of base from \(a\) to \(b\)	\(\log_a x = \dfrac{\log_b x}{\log_b a}\)

You may recognize natural logarithms from the integral of \(1/x\). This comes up very frequently in thermodynamics and kinetics. Here we can see the solution where we make use of the logarithm algebra property for subtraction.

\[\int_{x_1}^{x_2}\frac{dx}{x} = \left. \ln x \right|_{x_1}^{x_2} = \ln x_2 - \ln x_1 = \ln\frac{x_2}{x_1}\]