ELF.APP Applications of Logarithms and Exponential Functions Applications of logarithms and
exponential functions appear almost immediately after these functions are
introduced as early as beginning algebra,
The most frequent application of logarithms historically and before the advent of electronic
calculators was for scientific calculations and understanding the
operation of slide rules.
After calculators and computers made computation much easier, the
exponential and logarithm functions continue to have great importance
as functions for modeling growth and decay in many contexts,
Financial models entailing continuous compounding use the natural
exponential model; population models use simple exponential models as
well as more subtle logistic models; and the study of radioactive
decay and newton's law of cooling both lead to exponential
functions that model decreasing functions.
Logarithmic functions arise frequently as well, to solve equations
related to exponential functions and in the derivation of exponential
functions as
models from differential equations.
Visualizing the information used in these applications is sometimes
connected to Cartesian graphs or often ignored with only algebraic
treatments presented.
In this section examples are given to illustrate how mapping diagrams can contribute a visual approach to these applications.
Applications of these functions have three canonical problems:
Given the modeling function, $f$ that depends on time, $t$, find the valueat a particular time, $t*$, that is find $f(t*)$. [Evaluate the function]
Given the modeling function, $f$ that depends on time, $t$, and a value of that function, $v$, find any time, $t_v$, where $f(t_v) = v$. [Solve the equation]
Given an exponential form of a modeling function with undetermined
parameters, $f$ that depends on time, $t$, and some value(s) at
particular time (s), such as $f(t_1) = v_1$ and $f(t_2)= v_2$, find the
parameters of $f$ useful in solving the previous problems.
Example ELF.APP.1:
Continuously Compound Interest. $A(t)=Pe^{rt}$. $P=$ principal;
$r =$ rate of interest per annum as decimal; $t =$ time invested.
ExampleELF.APP.2:Radioactive Decay.$A(t)=A_0e^{rt}$.
$A_0=$ initial mass of radioactive isotope; $r =$ rate of decay
per period; $t =$ time elapsed from initial time.
ExampleELF.APP.3: Newton's Law of Cooling. $T(t)=T_R + \Delta T_0 e^{kt}$. $T_R=$ ambient (room) temperature; $ \Delta T_0 =$ initial temperature difference; $k =$ rate of temperature difference change; $t =$ time elapsed.
Example ELF.APP.4:
Simple Exponential Growth. $P(t)=P_0e^{rt}$. $P_0=$ initial population;
$r =$ rate of population growth; $t =$ time elapsed from initial
time.
ExampleELF.APP.5:A Logistic Population Model.$P(t)=\frac L{1+e^{-t}}$. $L=$ limit population; $t =$ time elapsed from initial time.