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 value at 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.
Example  ELF.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.

Example  ELF.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.
Example ELF.APP.5 :A Logistic Population Model.$P(t)=\frac L{1+e^{-t}}$. $L=$ limit population; $t =$ time elapsed from initial time.