5.3: The natural logarithm

Definition

The natural logarithm function, \(\ln\), is defined for \(u > 0\) by

\[\lambda=\ln u=\log _{e} u \Longleftrightarrow u=e^{\lambda} \label{5.8}\]

The natural logarithm is the logarithm to the base e and the properties of logarithms for all bases apply:

\[\ln (A \times B)=\ln (A)+\ln (B) \label{5.9}\]

\[\ln (A / B) =\ln (A)-\ln (B) \label{5.10}\]

\[\ln \left(A^{c}\right) =c \times \ln (A) \label{5.11}\]

\[\text { If } d>0, d \neq 1, \text { then } \quad \log _{d} A =\frac{\ln A}{\ln d} \label{5.12}\]

\[u =e^{\ln u} \label{5.13}\]

\[\ln \left(e^{\lambda}\right) =\lambda \label{5.14}\]

The natural logarithm is keyed on most calculators as ln or LN. Equation \ref{5.12} shows that all logarithms may be calculated using just the natural logarithm. However, \(\log _{10}\) is also keyed on most calculators and the key may be labeled Log or LOG. But in MATLAB log(x) means natural logarithm of x.

Using Equation \ref{5.13} we can express all exponential functions in terms of the base e. To see this, suppose \(B > 0\) and \(E(t) = B^t\). Equation \ref{5.13} states that

\[B=e^{\ln B}\]

so we may write \(E(t) = B^t\) as

\[E(t)=B^{t}=\left(e^{\ln B}\right)^{t}=e^{t \ln B}\]

For example, \(\ln 2 \doteq 0.631472\). For \(E(t) = 2^t\) we have

\[E(t)=2^{t}=\left(e^{\ln 2}\right)^{t}=e^{t \ln 2} \doteq e^{0.631472 t}\]

As a consequence, functions of the form \(E(t) = e ^{kt}\), \(k\) a constant, are very important and we compute their derivative in the next section.

Example 5.3.1 We found in Chapter 1 that cell density (measured by light absorbance, Abs) of Vibrio natriegens growing in a flask at pH 6.25, data in Table 1.1, was described by (Equation 1.1.17)

\[\mathrm{Abs}=0.0174 \times 1.032^{\mathrm{Time}}\]

Using natural logarithm, we write this in terms of e as

\[\begin{aligned}
\mathrm{Abs} &=0.0174 \times 1.032^{\text {Time }} &\\
\mathrm{Abs} &=0.0174 ~ \left[e^{\ln 1.032}\right]^{\text {Time }} & \text{Equation } 5.13\\
\mathrm{Abs} &=0.0174 ~ \left[e^{0.03150}\right]^{\text {Time }} & \ln 1.022 \doteq 0.02176\\
\mathrm{Abs} &=0.0174 ~ e^{0.03150 \times \text { Time }} & \text{Equation } 5.11\\
\end{aligned}\]

 

Exercises for Section 5.3, The natural logarithm.

Exercise 5.3.1 You should have found in Explore 1.6.1 that plasma penicillin during 20 minutes following injection of two grams of penicillin could be computed as

\[P(T)=200 \times 0.77^{T} \quad T=\text { index of five minute intervals. }\]

1. Write P in the form of \(P(T)=200 e^{-k_{1} T}\).
2. Write P in the form of \(P(t)=200 e^{-k_{2} t}\) where \(t\) measures time in minutes.

Exercise 5.3.2 Use Equation \ref{5.12}, \(\log _{d} A=\ln A / \ln d\) to compute \(\log _{2} A\) for \(A = 1, 2, 3, \cdots, 10\).

Exercise 5.3.3 We found in Section 1.3 that light intensity, \(I_d\), as a function of depth, \(d\) was given by

\[I_{d}=0.4 \times 0.82^{d}\]

Find \(k\) so that \(I_{d}=0.4 e^{k \times d}\).

Exercise 5.3.4 Write each of the following functions in the form \(f(t)=A e^{k t}\).

1. \(f(t)=5 \cdot 10^{t}\)
2. \(f(t)=5 \cdot 10^{-t}\)
3. \(f(t)=7 \cdot 2^{t}\)
4. \(f(t)=5 \cdot 2^{-t}\)
5. \(f(t)=5\left(\frac{1}{2}\right)^{t}\)
6. \(f(t)=5\left(\frac{1}{2}\right)^{-t}\)

Exercise 5.3.5 Use the Properties of Logarithms, Equations \ref{5.9} - \ref{5.14} to write each of the following functions in the form \(f(t) = A + B \ln {t}\).

1. \(f(t)=5 \log _{10} t\)
2. \(f(t)=5 \log _{2} t^{3}\)
3. \(f(t)=7 \log _{5} 5 t\)
4. \(f(t)=5 \log _{10} 3 t\)
5. \(f(t)=3 \log _{4}\left(t / 2^{3}\right)\)
6. \(f(t)=3 \log _{8}\left(16 t^{10} \right)\)